# Cluster expansion methods in rigorous statistical mechanics

Aldo Procacci

###### Contents

  1. I Classical Continuous Systems
    1. 1 Ensembles in Continuous systems
      1. 1.1 The Hamiltonian and the equations of motion
      2. 1.2  Gibbsian ensembles
      3. 1.3 The Micro-Canonical ensemble
      4. 1.4 The entropy is additive. An euristic discussion
      5. 1.5 Entropy of the ideal gas
      6. 1.6  The Gibbs paradox
      7. 1.7  The Canonical Ensemble
      8. 1.8 Canonical Ensemble: Energy fluctuations
      9. 1.9 The Grand Canonical Ensemble
      10. 1.10 The ideal gas in the Grand Canonical Ensemble
      11. 1.11 The Thermodynamic limit
    2. 2 The Grand Canonical Ensemble
      1. 2.1 Conditions on the potential energy
      2. 2.2 Potentials too attractive at short distances
      3. 2.3  Two examples of infrared catastrophe
        1. 2.3.1 Potential with hard core too attractive at large distances
        2. 2.3.2 Potential with hard core too repuslive at large distances
      4. 2.4 The Ruelle example
      5. 2.5 Admissible potentials
        1. 2.5.1 Basuev Criteria
      6. 2.6 The infinite volume limit
      7. 2.7 Example: finite range potentials
      8. 2.8 Properties the pressure
      9. 2.9 Continuity of the pressure
      10. 2.10 Analiticity of the pressure
    3. 3 High temperature low density expansion
      1. 3.1 The Mayer series
        1. 3.1.1 Some Notations about graphs
        2. 3.1.2 Mayer Series: definition
        3. 3.1.3 The combinatorial problem
      2. 3.2 Convergence of the Mayer series
        1. 3.2.1 Kirkwood-Salsburg equations: the original proof of Theorem 3.2 (according to Penrose)
      3. 3.3 The Penrose Tree graph Identity
        1. 3.3.1 The original Penrose map
      4. 3.4 The hard-sphere gas via Penrose identity
      5. 3.5 Stable and tempered potentials
      6. 3.6 Convergence of the Virial expansion
  2. II Discrete systems
    1. 4 The polymer gas
      1. 4.1 Setting
      2. 4.2 Convergence of the abstract polymer gas
        1. 4.2.1 Reorganization of the series $\Pi^{*}_{\gamma_{0}}(\bm{\rho})$
        2. 4.2.2 Trees and convergence
        3. 4.2.3 Convergence criteria
        4. 4.2.4 Elementary Examples
      3. 4.3 Gas of non-overlapping subsets
        1. 4.3.1 Convergence via the abstract polymer criteria
        2. 4.3.2 The subset gas: proof by Induction
    2. 5 Two systems in the cubic lattice
      1. 5.1 The Self repuslive Lattice gas
        1. 5.1.1 Covergence by direct Mayer expansion
        2. 5.1.2 High temperature polymer expansion of the lattice gas
      2. 5.2 The BEG model at low temperature
    3. 6 Spin systems in a lattice
      1. 6.1 Some examples
      2. 6.2 Polymer gas representation of $Z_{\Lambda}(\beta)$
        1. 6.2.1 General spin systems interacting via pair potentials
        2. 6.2.2 Polymer representation of $Z_{\Lambda}(\beta)$ for the pair potential case
        3. 6.2.3 The bounded spin system case
        4. 6.2.4 The unbounded spin system case
      3. 6.3 Many-body interactions
      4. 6.4 Some applications of theorem 6.4
        1. 6.4.1 The Ising model
        2. 6.4.2 Direct proof of the Israel theorem
        3. 6.4.3 The lattice gas with $N$ body interactions
    4. 7 Ising model
      1. 7.0.1 High temperature expansion
      2. 7.0.2 Low temperature expansion
      3. 7.0.3 Existence of phase transitions
      4. 7.0.4 The critical temperature

##  Part I Classical Continuous Systems

###  Chapter 1 Ensembles in Continuous systems

####  1.1 The Hamiltonian and the equations of motion

In this first part of the book we will deal with a system made by a large
number $N$ of continuous particles enclosed in a box
$\Lambda\subset\mathbb{R}^{d}$ (we will assume $\Lambda$ to be cube and denote
$|\Lambda|$ its volume) performing a motion according to the laws of classical
mechanics. “Large number of particles” in physics lingo means typically
$N\approx 10^{23}$.

We will restrict our discussion to systems composed of identical particles
with no internal structure, i.e. just “point” particles with a given mass $m$.
The position of the $i^{th}$ particle in the box $\Lambda$ at a given time $t$
is given by a $d$-component coordinate vector, denoted by ${x_{i}}=x_{i}(t)$
with respect to some system of orthogonal axis.

The momentum of the $i^{th}$ particle at time $t$ is also given by a
$d$-component vector denoted by ${p_{i}}=p_{i}(t)$ which is directly related
to the velocity of the particles, i.e. if $m$ is the mass of the particle,
then $p_{i}=m(dx_{i}/dt)$.

In principle, the laws of mechanics permit to know the evolution of such a
system in time, i.e. by these laws one should be able to determine which
positions $x_{i}=x_{i}(t)$ and momenta $p_{i}=p_{i}(t)$ the particles in the
system will have in the future and had in the past, provided one knows the
position of the particles $x^{0}_{i}=x_{i}(t_{0})$ and the momenta of
$p^{0}_{i}=p_{i}(t_{0})$ at a given time $t_{0}$.

As a matter of fact, the time evolution of such system is described by a real
valued function $H(p_{1},\dots,p_{N},x_{1},\dots,x_{N})$ of particle positions
and momenta (hence a function of $2dN$ real variables) called the Hamiltonian.
In the case of isolated systems, this function $H$ is assumed to have the form

$H(p_{1},\dots,p_{N},x_{1},\dots,x_{N})=\sum_{i=1}^{N}{p_{i}^{2}\over
2m}+U(x_{1},\dots x_{N})$ $None$

The term $\sum_{i=1}^{N}{p_{i}^{2}\over 2m}$ is called the kinetic energy of
the system, while the term $U(x_{1},\dots x_{N})$ is the potential energy.
Since we are assuming that particles have to be enclosed in a box $\Lambda$ we
still have to restrict $x_{i}\in\Lambda$ for all $i=1,2,\dots,N$, while no
restriction is imposed on $p_{i}$, i.e. the momentum (and hence the velocity)
of particles can be arbitrarily large.

Once the Hamiltonian of the system is given, one could in principle solve the
system of $2dN$ differential equations

$\begin{aligned} {dx_{i}\over dt}&={\partial H\over\partial p_{i}}\\\
{dp_{i}\over dt}&=-{\partial H\over\partial
x_{i}}\end{aligned}~{}~{}~{}~{}~{}~{}~{}~{}~{}i=1,\dots,N$ $None$

where ${\partial/\partial x_{i}}$ and ${\partial/\partial p_{i}}$ are
$d$-dimensional gradients.

This is a system of $2dN$ first order differential equations. The solution are
$2dN$ functions $x_{i}(t)$, $p_{i}(t)$, $i=1,\dots,N$, given the positions
$x_{i}(t_{0})$ and momenta $p_{i}(t_{0})$ at some initial time $t=t_{0}$.

It is convenient to introduce a $2dN$-dimensional space $\Gamma_{N}(\Lambda)$,
called the phase space of the system (with $N$ particles) whose points are
determined by the coordinates $(\bm{q},\bm{p})$ with
$\bm{q}=(x_{1},\dots,x_{N})$ and $\bm{p}=(p_{1},\dots,p_{N})$ ($\bm{q}$ and
$\bm{p}$ are both $dN$ vectors!) with the further condition that
$x_{i}\in\Lambda$ for all $i=1,2,\dots,N$. A point $(\bm{q},\bm{p})$ in the
phase space of the system is called a microstate of the system. With these
notations the evolution of the system during time can be interpreted as the
evolution of a point in the “plane” $\bm{q},\bm{p}$ (actually a $2dN$
dimension space, the phase space $\Gamma_{N}(\Lambda)$).

We finally want to remark that, by (1.1) (isolated system), the value of the
Hamiltonian $H$ is constant during the time evolution of the system governed
by the Hamilton equations (1.2). Namely
$H(\bm{q}(t),\bm{p}(t))=H(\bm{q}(0),\bm{p}(0))=E$. To see this, just calculate
the total derivative of $H$ respect to the time using equations (1.2). This
constant $E$ of motion is called energy of the system. Thus the trajectory of
the point $\bm{q},\bm{p}$ in the phase space $\Gamma_{N}(\Lambda)$ occurs in
the surface $H(\bm{p},\bm{q})=E$.

####  1.2  Gibbsian ensembles

It is substantially meaningless to look for the solution of system (1.2).

First, there is a technical reason. Namely, the system contains an enormous
number of equations ($\approx 10^{23}$) which in general are coupled
(depending on the structure of $U$), so that it is practically an impossible
task to find the solution.

However, even supposing that some very powerful entity would give us the
solution, this extremely detailed description (a microscopic description)
would not be useful to describe the macroscopic properties of such a system.
Macroscopic properties which appear to us as the laws of thermodynamic are due
presumably by some mean effects of such large systems and can in general be
described in terms of very few parameters, e.g. temperature, volume, pressure,
etc.. Hence we have no means and also no desire to know the microscopic state
of the system at every instant (i.e. to know the functions
$\bm{q}(t),\bm{p}(t)$). We thus shall adopt a statistical point of view in
order to describe the system.

We know a priori some macroscopic properties of the system, e.g. an isolated
system occupies the volume $\Lambda$, has $N$ particles and has a fixed energy
$E$.

We further know that macroscopic systems, if not perturbated from the
exterior, tend to stay in a situation of macroscopic (or thermodynamic)
equilibrium (i.e. a “static” situation), in which the values of some
thermodynamic parameters (e.g. pressure, temperature, etc.) are well defined
and fixed and do not change in time. Of course, in a system at the
thermodynamic equilibrium, the situation at the microscopic level is
desperately far form a static one. Particles in a gas at the equilibrium do in
general complicated and crazy motions all the time, nevertheless nothing seems
to happen as time goes by at the macroscopic level. So the thermodynamic
equilibrium of a system must be the effect of some mean behavior at the
microscopic level. This “static” mean macroscopic behavior of systems composed
by a large number of particles must be produced in some way by the microscopic
interactions between particles and by the law of mechanics.

Adopting the statistical mechanics point of view to describe a macroscopic
system at equilibrium means that we renounce to understand how and why a
system reach the thermodynamic equilibrium starting from the microscopic
level, and we just assume that, at the thermodynamic equilibrium
(characterized by some thermodynamic parameters), the system could be found in
any microscopic state within a certain suitable set of microstates compatible
with the fixed thermodynamic parameters. In other words, over long periods,
the time spent by a system in some region of the phase space of microstates
with the same energy is proportional to the volume of this region, i.e., that
all accessible microstates are equiprobable over a long period of time. This
is the so called ergodic hypothesis. Of course, in order to have some hope
that such a point of view will work, we need to treat really “macroscopic
systems”. So values such $N$ and $V$ must be always thought as very large
values (i.e. close to $\infty$).

The statistical description of the macroscopic properties of the system at
equilibrium (and in particular the laws of thermodynamic) is done in two
steps. Step 1: fixing the Gibbsian ensemble (or the space of configurations).
We choose the phase space $\Gamma_{e}$ and we assume that the system can be
found in any microstate $(\bm{q},\bm{p})\in\Gamma_{e}$. This set $\Gamma_{e}$
has to interpreted as the set of all microstates accessible by the system and
it is called the Gibbsian ensemble or the space of configurations of the
system. We will see later that several choices are possible for $\Gamma_{e}$.
We will then think not on a single system, but in an infinite number of mental
copies of the same system, one copy for each element of $\Gamma_{e}$.

Step 2: fixing the Gibbs measure in the Gibbisan ensemble.

We choose a function $\rho(\bm{p},\bm{q})$ in $\Gamma_{e}$ which will
represent the probability density in the Gibbsian ensemble. Namely,
$\rho(\bm{p},\bm{q})$ is a function such that

$\int_{\Gamma_{e}}\rho(\bm{p},\bm{q})d\bm{p}d\bm{q}=1$

and $d\mu(\bm{p},\bm{q})=\rho(\bm{p},\bm{q})d\bm{p}d\bm{q}$ represents the
probability to find the system in a microstate (or in the configuration)
contained in an infinitesimal volume $d\bm{p}d\bm{q}$ around the point
$(\bm{p},\bm{q})\in\Gamma_{e}$ where $d\bm{p}~{}d\bm{q}$ is the usual Lebesgue
measure in $\mathbb{R}^{2dN}$. The measure $\mu(\bm{p},\bm{q})$ defined in
$\Gamma_{e}$ is called the Gibbs measure of the system.

Once a Gibbsian ensemble and a Gibbs measure are established, one can begin to
do statistic in order to describe the macroscopic state of the system. When we
look at a macroscopic system described via certain Gibbs ensemble we do not
know in which microstate the system is at a given instant. All we know is that
its microscopic state must be one of the microstates of the space
configuration $\Gamma_{e}$ with probability density given by the Gibbs measure
$d\mu$.

For example, suppose that $f(\bm{p},\bm{q})$ is a measurable function respect
to the Gibbs measure $d\mu$, such as the energy, the kinetic energy per
particle, potential energy etc. Then we can calculate its mean value in the
Gibbsian ensemble that we have chosen by the formula:

$\langle f\rangle~{}=~{}\int
f(\bm{p},\bm{q})\rho(\bm{p},\bm{q})d\bm{p}d\bm{q}$

We also recall the concept of mean relative square fluctuation of $f$ (a.k.a.
standard deviation) denoted by $\sigma_{f}$. This quantity measures how spread
is the probability distribution of $f(\bm{p},\bm{q})$ around its mean value.
It is defined as

$\sigma_{f}~{}~{}=~{}~{}{\langle(f-\langle f\rangle)^{2}\rangle\over\langle
f\rangle^{2}}~{}~{}=~{}~{}{\langle f^{2}\rangle-\langle
f\rangle^{2}\over\langle f\rangle^{2}}$ $None$

####  1.3 The Micro-Canonical ensemble

There are different possible choices for the ensembles, depending on the
different macroscopic situation of the system.

We start defining the micro-Canonical ensemble which is used to describe
perfectly isolated systems. Hence we suppose that our system is totally
isolated from the outside, the $N$ particles are constrained to stay in the
box $\Lambda$, and they do not exchange energy with the outside, so that the
system has a fixed energy $E$, occupies a fixed volume $|\Lambda|$ and has a
fixed number of particles $N$.

Thus, for such a system, we can naturally say that the space of configuration
$\Gamma_{mc}$ is the set of points $\bm{p},\bm{q}$ (with
$\bm{p}\in\mathbb{R}^{dN}$ and $\bm{q}\in\Lambda^{N}$) with energy between a
given value $E$ and $E+\Delta E$ (where $\Delta E$ can be interpreted as the
experimental error in the measure of the energy $E$). We have

$\Gamma_{mc}(E,\Lambda,N)~{}=~{}\\{(\bm{p},\bm{q}):\bm{p}\in\mathbb{R}^{dN},~{}\bm{q}\in\Lambda^{N}~{}{\rm
and}~{}E<H(\bm{p},\bm{q})<E+\Delta E\\}$

We now choose the probability measure in such way that any microstate in the
set of configurations above is equally probable, i.e. there is no reason to
assign different probabilities to different microstates. This quite drastic
hypothesis is the so-called postulate of equal a priori distribution, which is
just a different formulation of the ergodic hypothesis. Hence

$\rho(\bm{p},\bm{q})~{}~{}=~{}~{}\begin{cases}~{}[\Psi_{\Lambda}(E,N)]^{-1}&~{}{\rm
if}~{}(\bm{q},\bm{p})\in\Gamma_{mc}\\\ 0&~{}{\rm otherwise}\end{cases}$

where

$\Psi_{\Lambda}(E,N)~{}=~{}\int_{E<H(\bm{p},\bm{q})<E+\Delta
E}d\bm{p}\,d\bm{q}$ $None$

is the normalization constant, which in this case coincides with the
$2dN$-dimensional “volume” in the phase space occupied by the space of
configuration of the micro-canonical ensemble. $\Psi_{\Lambda}(E,N)$ is
generally called the partition function of the system in the Micro-Canonical
ensemble. The link between the Micro-Canonical ensemble and the thermodynamic
is obtained via the definition of the thermodynamic entropy of the system by

$S_{\Lambda}(E,N)~{}~{}=~{}~{}k\ln\left[{1\over|\delta|}\Psi_{\Lambda}(E,N)\right]$
$None$

where $k$ is the Boltzmann’s constant and $|\delta|$ is the volume of some
elementary phase cell $\delta$ in the phase space, so that the pure number
${|\delta|}^{-1}\Psi_{\Lambda}(E,N)$ is the number of such cells in the
configuration space.

It may seem that the value of this constant $|\delta|$ can be somewhat
arbitrary, since it depends on our measure instruments, and it could be done
as small as we please by improving our measure techniques. But the
experimental measures tell us that this constant must be fixed at the value
$h^{dN}$ where $h$ is the Plank constant. So hereafter we will assume, unless
differently specificated, that $|\delta|$ is set at the value of the Plank
constant. It is important to stress that the presence of this constant in the
definition of the entropy has a very deep physical meaning. Actually, it is a
first clue of the quantum mechanics nature of particle systems: things go as
if the position $\bm{p},\bm{q}$ of a micro-state in the phase space could not
be known exactly and one can just say that the micro-state is in a small cube
$d\bm{p}d\bm{q}$ centered at $\bm{p},\bm{q}$ of the space phase with volume
$h^{dN}$. This is actually the Heisenberg indetermination principle.

The definition (1.5) gives a beautiful probabilistic interpretation of the
second law of thermodynamic. A macroscopic system at the equilibrium will tend
to stay in a state of maximal entropy, namely, by (1.5), in the most probable
macroscopic state, i.e. a macroscopic state with thermodynamics parameters
fixed in such way that this state corresponds to the largest number of
microstates. Namely, at equilibrium, the quantity $\Psi_{\Lambda}(E,N)$ should
expect to reach a maximum value. Thus the entropy (which by definition is just
the logarithm of the number of micro-states of a macro-state) is also expected
to be maximum at equilibrium. So the second law of thermodynamics stating that
the entropy of an isolated system always increases means in term of
statistical mechanics that the systems tends to evolve to macrostates which
are more probable, i.e. those with maximum number of microstates.

It is also interesting to check that entropy (1.5) is an “extensive” quantity.
Namely, if the macroscopic system is composed by two macroscopic subsystems
whose entropies are, respectively $S_{1}$ and $S_{2}$, the entropy of the
total system must be $S_{1}+S_{2}$.

####  1.4 The entropy is additive. An euristic discussion

Suppose thus to consider a system made with two subsystems, one living in a
phase space $\Gamma_{1}$ with coordinates $\bm{p}_{1},\bm{q}_{1}$ occupying
the volume $\Lambda_{1}$ and described by the Hamiltonian
$H_{1}(\bm{p}_{1},\bm{q}_{1})$ and the other in a phase space $\Gamma_{2}$
with coordinates $\bm{p}_{2},\bm{q}_{2}$, occupying the volume $\Lambda_{2}$
and described by the Hamiltonian $H_{2}(\bm{p}_{2},\bm{q}_{2})$. We also
suppose that systems are isolated from each other. Consider first the micro-
canonical ensemble for each subsystem taken alone. The energy of the first
system will stay in an interval say $(E_{1},E_{1}+\Delta)$ while the second
system will have an energy in $(E_{2},E_{2}+\Delta)$. The entropies of the
subsystems will be respectively $S(E_{1})=k\ln\Psi_{1}(E_{1})$ and
$S(E_{2})=k\ln\Psi_{2}(E_{2})$, where $\Psi_{1}(E_{1})$ and $\Psi_{2}(E_{2})$
are the volumes occupied by the two ensembles in their respective phase spaces
$\Gamma_{1}$ and $\Gamma_{2}$. Consider now the micro-canonical ensemble of
the total system made by the two subsystems, The composite system lives in a
phase space $\Gamma_{1}\times\Gamma_{2}$ with coordinates
$p_{1},q_{1},p_{2},q_{2}$ occupying the volume $V_{1}+V_{2}$ and it is
described by the Hamiltonian $H_{1}(p_{1},q_{1})+H_{2}(p_{2},q_{2})$ (system
are supposed isolated one from each other). Let the total energy be in the
interval say $(E,E+2\Delta)$ ($\Delta\ll E$). This ensemble contains all the
micro-states of the composite system such that:

a) $N_{1}$ particles with momenta and coordinates $p_{1},q_{1}$ are in the
volume $V_{1}$

b) $N_{2}$ particles with momenta and coordinates $p_{2},q_{2}$ are in the
volume $V_{2}$

c) The energy $E_{1}$ and $E_{2}$ of the subsystems have values satisfying the
condition

$E<E_{1}+E_{2}<E+2\Delta$ $None$

We want to calculate the partition function $\Psi(E)$ of the composite system.
Clearly $\Psi_{1}(E_{1})\Psi_{2}(E_{2})$ is the volume in the composite phase
space $\Gamma$ with coordinate $(\bm{p}_{1},\bm{p}_{2},\bm{q}_{1},\bm{q}_{2})$
that corresponds to conditions a) and b) with first system at energy $E_{1}$
and second system at energy $E_{2}$ such that $E_{1}+E_{2}\in(E,E+2\Delta)$.
Then

$\Psi(E)~{}=~{}\sum_{E_{1},E_{2}\atop
E<E_{1}+E_{2}<E+2\Delta}\Psi_{1}(E_{1})\Psi_{2}(E_{2})$

Since $E_{1}$ and $E_{2}$ are possible values of $H(\bm{p}_{1},\bm{q}_{1})$
and $H(\bm{p}_{2},\bm{q}_{2})$ , suppose that $H(p_{1},q_{1})$ and
$H(\bm{p}_{2},\bm{q}_{2})$ are bounded below (as it will be always the case,
see later) and for simplicity let the joint lower bound be equal to $0$. Hence
$E_{1}$ and $E_{2}$ both varies in the interval $[0,E]$. Suppose also that
$E_{1}$ and $E_{2}$ take discrete values $E_{i}=0,\Delta,2\Delta,...$ so that
in the interval $(0,E)$ there are $E/\Delta$ of such intervals. Then

$\Psi(E)~{}=~{}\sum_{i=1}^{E/\Delta}\Psi_{1}(E_{i})\Psi_{2}(E-E_{i})$ $None$

The entropy of the total system of $N=N_{1}+N_{2}$ particles, volume
$\Lambda~{}=~{}\Lambda_{1}\cup\Lambda_{2}$ and energy $E$ is given by

$S_{\Lambda}(E,N)~{}~{}=~{}~{}k\ln\left[\sum_{i=1}^{E/\Delta}\Psi_{1}(E_{i})\Psi_{2}(E-E_{i})\right]$

As subsystems are supposed macroscopic ($N_{1}\to\infty$ and $N_{2}\to\infty$)
is easy to see that a single term in the sum (1.7) will dominate. Sum (1.7) is
a sum of positive terms, let the largest of such terms be
$\Psi_{1}(\bar{E}_{1})\Psi_{2}(\bar{E}_{2})$ with
$\bar{E}_{1}+\bar{E}_{2}~{}=~{}E$. Then we have the obvious inequalities

$\Psi_{1}(\bar{E}_{1})\Psi_{2}(\bar{E}_{2})~{}\leq~{}\Psi(E)~{}\leq~{}{E\over\Delta}\Psi_{1}(\bar{E}_{1})\Psi_{2}(\bar{E}_{2})$

or

$k\ln\left[\Psi_{1}(\bar{E}_{1})\Psi_{2}(\bar{E}_{2})\right]~{}\leq~{}S_{\Lambda}(E,N)~{}\leq~{}k\ln\left[\Psi_{1}(\bar{E}_{1})\Psi_{2}(\bar{E}_{2})\right]+k\ln(E/\Delta)$
$None$

We expect, as $N_{1}\to\infty$ and $N_{2}\to\infty$, that $\Psi_{1}\sim
C^{N_{1}}$ and $\Psi_{2}\sim C^{N_{2}}$, thus $\ln\Psi_{1}\propto N_{1}$ and
$\ln Z_{2}\propto N_{2}$. and also $E\sim N_{1}+N_{2}$. Hence factor
$\ln(E/\Delta)$ goes like $\ln N$ and can be neglected. Namely by this
discussion (just a counting argument) we get

$S_{\Lambda}(E,N)~{}=~{}S_{\Lambda_{1}}(\bar{E}_{1},N_{1})+S_{\Lambda_{2}}(\bar{E}_{2},N_{2})+O(\ln
N)$ $None$

In other words the entropy is extensive, modulo terms of order $\ln N$. Note
that (1.9) also means that the two subsystems has a definite values
$\bar{E}_{1}$ and $\bar{E}_{2}$ for the energy. Namely $\bar{E}_{1}$ and
$\bar{E}_{2}$ are the values that maximize the number

$\Psi_{1}(E_{1})\Psi_{2}(E_{2})$

under the condition $E_{1}+E_{2}~{}=~{}E$. Using Lagrange multiplier is easy
to check that

$\left.{\partial\ln\Psi_{1}(E_{1})\over\partial
E_{1}}\right|_{E_{1}=\bar{E}_{1}}~{}=~{}\left.{\partial\ln\Psi_{2}(E_{2})\over\partial
E_{2}}\right|_{E_{2}=\bar{E}_{2}}$

or

$\left.{\partial S(E_{1})\over\partial
E_{1}}\right|_{E_{1}=\bar{E}_{1}}~{}=~{}\left.{\partial S(E_{2})\over\partial
E_{2}}\right|_{E_{2}=\bar{E}_{2}}$

Since thermodynamics tells us that ${\partial S(E,V)\over\partial
E}~{}=~{}{1\over T}$, we conclude that the two subsystems choose energy
$\bar{E}_{1}$ and $\bar{E}_{2}$ in such way to have the same temperature. Thus
the temperature in a macroscopic system can be seen as the parameter governing
the equilibrium between one part of the system and the other.

####  1.5 Entropy of the ideal gas

If one can calculate the partition function of the Micro-canonical ensemble,
then it is possible to derive the thermodynamic properties of the system. The
Micro-Canonical ensemble is difficult to be treated mathematically. As a
matter of fact, a direct calculation of the integral in r.h.s. of (1.4) is
generally very difficult, since involves integration over complicated surfaces
in high dimensions. According to elementary calculus in $\mathbb{R}^{n}$ we
can use a volume integral to calculate the integral (1.4). Volume integrals
are easier to deal with than surface integrals. Let $\omega_{\Lambda}(E,N)$ be
the volume of the phase space surrounded by the surface $H(\bm{p},\bm{q})=E$
(with of course the further condition that particles are constrained to stay
in $\Lambda$). Then

$\omega_{\Lambda}(E,N)~{}=~{}\int_{H(\bm{p},\bm{q})\leq E}d\bm{p}~{}d\bm{q}$
$None$

Then, for small $\Delta E$

$\Psi_{\Lambda}(E,N)~{}=~{}\omega_{\Lambda}(E+\Delta
E,N)-\omega_{\Lambda}(E,N)\approx{\partial\omega_{\Lambda}(E,N)\over\partial
E}\Delta E$ $None$

Let for example calculate the Micro-Canonical partition function of an ideal
gas, i.e. a gas of non interacting particles. with Hamiltonian

$H~{}=~{}\sum_{i=1}^{N}{p_{i}^{2}\over 2m}$ $None$

We thus calculate $\omega(N,\Lambda,E)$ when $H(\bm{p},\bm{q})$ is given by
(1.12). In this case it is absolutely elementary to calculate the integral
$\int_{\Lambda}dq$ which gives just $|\Lambda|$ i.e. the volume occupied by
$\Lambda$, and therefore we get

$\displaystyle\omega_{\Lambda}(E,N)$ $\displaystyle=\int_{H(\bm{p},\bm{q})\leq
E}d\bm{p}~{}d\bm{q}$
$\displaystyle=|\Lambda|^{N}\int_{\sum_{i=1}^{N}{p_{i}^{2}\over 2m}\leq
E}dp_{1}\dots dp_{N}$
$\displaystyle=|\Lambda|^{N}\int_{\sum_{i=1}^{N}{p_{i}^{2}}\leq
2mE}dp_{1}\dots dp_{N}$

the last integral is just the volume of a $dN$ dimensional sphere of radius
$\sqrt{2mE}$. Let us thus face this geometric problem. The volume of a sphere
of given radius $R$ in a $n$ dimensional space is the integral

$V_{n}(R)~{}=~{}\int_{\sum_{i=1}^{n}x_{1}^{2}\leq R^{2}}dx_{1}\dots
dx_{n}~{}=~{}C_{n}R^{n}$

In order to find $C_{n}$ consider the following integral

$\displaystyle\int_{\mathbb{R}^{n}}e^{-(x_{1}^{2}+\dots+x_{n}^{2})}dx_{1}\dots
dx_{n}$
$\displaystyle=\int_{-\infty}^{+\infty}dx_{1}\dots\int_{-\infty}^{+\infty}dx_{n}e^{-(x_{1}^{2}+\dots+x_{n}^{2})}$
$\displaystyle=\left(\int_{-\infty}^{+\infty}dxe^{-x^{2}}\right)^{n}$
$\displaystyle=\pi^{n\over 2}$

on the other hand, noting that the integrand above depends only on
$r=(x_{1}^{2}+\dots x_{n}^{2})^{1/2}$, by a transformation to polar
coordinates in $n$ dimensions we can express the volume element $dx_{1}\dots
dx_{n}$ by spherical shells $dV_{n}(r)=dC_{n}r^{n}=nC_{n}r^{n-1}dr$. Then
integral above can also be calculated as

$\displaystyle\int_{\mathbb{R}^{n}}e^{-(x_{1}^{2}+\dots+x_{n}^{2})}dx_{1}\dots
dx_{n}$ $\displaystyle=\int_{0}^{\infty}e^{-r^{2}}dV_{n}(r)$ $None$
$\displaystyle=nC_{n}\int_{0}^{\infty}r^{n-1}e^{-r^{2}}dr$
$\displaystyle={nC_{n}\over 2}\int_{0}^{\infty}x^{n/2-1}e^{-x}dx$

Recall now the definition of the gamma function: for any $z>0$

$\Gamma(z)~{}~{}=~{}~{}\int_{0}^{\infty}x^{z-1}e^{-x}dx$

Among properties of the gamma function we recall

$\Gamma(n)~{}=~{}(n-1)!~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm
n~{}positive~{}integer}$

and

$z\Gamma(z)~{}=~{}\Gamma(z+1)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm
z\in\mathbb{R}^{+}}$

So Gamma function is an extension of factorial in the whole positive real
axis.

Equation (1.13) thus becomes

${nC_{n}\over 2}\Gamma({n/2})~{}=~{}\pi^{n/2}$

and consequently of the volume $V_{n}(R)$ of a sphere on radius $R$ in $n$
dimensions.

$C_{n}~{}=~{}{\pi^{n/2}\over\Gamma({n\over
2}+1)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}V_{n}(R)~{}=~{}{\pi^{n/2}\over\Gamma({n\over
2}+1)}R^{n}$

Hence we get

$\omega_{\Lambda}(E,N)~{}=~{}{\pi^{3N/2}\over{3N\over 2}\Gamma({3N\over
2})}(2mE)^{3N\over 2}|\Lambda|^{N}$

and, by (1.11)

$\Psi_{\Lambda}(E,N)~{}=~{}{\partial\omega_{\Lambda}(E,N)\over\partial
E}\Delta E~{}=~{}\Delta E~{}|\Lambda|^{N}~{}{\pi^{3N/2}\over\Gamma({3N\over
2})}(2m)^{3N/2}E^{{3N\over 2}-1}$

So, recalling definition (1.5), the entropy of an ideal gas is given by

$S_{\Lambda}(E,N)~{}=~{}k\ln\left[{\Delta
E}~{}|\Lambda|^{N}~{}{\pi^{3N/2}\over\Gamma({3N\over
2})}(2m/h^{2})^{3N/2}E^{{{3N\over 2}-1}}\right]$

Since $N$ is a very big number, we may write, for $N\gg 1$,

$E^{{3N\over 2}-1}\approx E^{3N\over
2},~{}~{}~{}~{}\ln\left[\Gamma\left({3N\over 2}\right)\right]\approx{3N\over
2}\left(\ln(3N/2)-1\right),~{}~{}~{}~{}k\ln(\Delta E)=O(1)\approx 0$

we used the Stirling approximation for the factorial: $n!\approx{n^{n}\over
e^{n}}$ for large $n$) thus

$S_{\Lambda}(E,N)~{}=~{}kN\left\\{{3\over
2}+\ln\left[~{}|\Lambda|~{}\left({4m\pi E\over
3Nh^{2}}\right)^{3/2}\right]\right\\}$ $None$

This equation leads to the correct equation of state for a perfect gas. In
fact, according to the laws of thermodynamics, the inverse temperature of the
system is the derivative of the entropy respect to the energy, and the
pressure is the derivative of the entropy respect to the volume $V=|\Lambda|$
times the temperature, i.e.

${1\over T}~{}=~{}{\partial S\over\partial E}~{}=~{}{3\over 2}{Nk\over
E}~{}~{}~{}~{}~{}~{}{\rm or}~{}~{}~{}~{}~{}~{}E~{}=~{}{3N\over 2}kT$ ${p\over
T}~{}=~{}{\partial S\over\partial V}~{}=~{}{Nk\over V}~{}~{}~{}~{}~{}~{}{\rm
or}~{}~{}~{}~{}~{}~{}PV~{}=~{}NkT$ $None$

Nevertheless (1.14) cannot be the correct expression for the entropy of an
ideal gas. One can just observe that l.h.s. of (1.14) is not a purely
extensive quantity, as the thermodynamic entropy should be. There is some deep
mistake in the calculation of the entropy. Indeed, by accepting that (1.14) is
the the correct expression of the thermodynamic entropy, one obtains the
called Gibbs paradox: an ideal gas confined in a box consisting initially of
two adjacent volumes $V_{A}$ and $V_{B}$ separated by a removable wall at the
thermodynamic equilibrium has entropy $S$ (calulated via (1.14)) and when the
wall is removed the entropy $S$ increases.

####  1.6  The Gibbs paradox

In this section we set $V=|\Lambda|$. Consider thus the entropy of an ideal
gas as a function of the temperature $T$, the volume $V$ and the numer of
particles $N$. By (1.14) and (1.15) we get

$S(T,V,N)~{}=~{}kN\left\\{{3\over 2}+\ln\left[~{}V~{}\left({2m\pi kT\over
h^{2}}\right)^{3/2}\right]\right\\}$ $None$

where, using (1.15), we have posed that $E/N={3\over 2}kT$. Consider now a
closed system consisting initially of two adjacent volumes $V_{A}$ and $V_{B}$
separated by a wall. The volume $A$ contains an ideal gas with $N_{A}$
particles, and the volume $V_{B}$ contains another ideal gas with $N_{B}$
particles. The two subsystems are kept at the same pressure $P$ and at the
same temperature $T$. The entropy of such system, according to (1.16) is

$S^{i}_{\rm total}~{}=~{}S(T,V_{A},N_{A})+S(T,V_{B},N_{B})$

If we now remove the wall, the two ideal gases will mix, each occupying the
volume $V_{A}+V_{B}$. In the new equilibrium situation the entropy is now

$S^{f}_{\rm total}~{}=~{}S(T,V_{A}+V_{B},N_{A})+S(T,V_{A}+V_{B},N_{B})$

The entropy difference between the initial state $i$ and the final state $f$
is, according to (1.16)

$\Delta S~{}=~{}S^{f}_{\rm total}-S^{i}_{\rm
total}~{}=~{}N_{A}k\ln(1+V_{B}/V_{A})+N_{B}k\ln(1+V_{A}/V_{B})$ $None$

So far everything seems to be fine, since $\Delta S>0$ as one should expect
for this irreversible process (the mixing on two ideal gases). But let us now
suppose that the two ideal gases are actually identical. We could repeat the
argument and again we will find that the variation of the entropy is $\Delta
S>0$. However this cannot be correct since, after the removal of the wall, no
macroscopic changes happens at all in the system. We could put back the wall
ad we will return to the initial macroscopic situation. This paradox is
clearly related to the fact that we are assuming the particles of the ideal
gas as distinguishable. If particle are considered as distinguishable, then
also the situation in which two identical perfect gases mixes is a
irreversible process. If we put back the wall particles in the volume $1$ are
not the same particles of the initial situation. The paradox was resolved by
Gibbs supposing that identical particles are not distinguishable. With this
hypothesis the number of microstates involving $N$ particles should be reduced
by a factor $N!$, since there are exactly $N!$ ways to enumerate $N$ identical
particles. Hence the correct definition of, e,g. $\omega_{\Lambda}(E,N)$
should be

$\omega_{\Lambda}(E,N)~{}=~{}{1\over N!}\int_{H(\bm{p},\bm{q})\leq E}dp~{}dq$
$None$

Thus, instead of (1.14), the correct entropy of a perfect gas is

$S_{\Lambda}(E,N)~{}=~{}kN\left\\{{3\over
2}+\ln\left[~{}|\Lambda|~{}\left({2m\pi kT\over
h^{2}}\right)^{3/2}\right]\right\\}-k\ln N!$ $None$

First observe that this new definition of the entropy does not affect the
equation of state of the perfect gas, since it differs from (1.14) by a term
independent on $E$ and $V$, so (1.19) leads to the same equations (1.15). and,
for $N\gg 1$, by Stirling’s formula $\ln N!\approx N\ln N-N$

$S_{\Lambda}(E,N)~{}=~{}kN\left\\{{5\over 2}+\ln\left[~{}{|\Lambda|\over
N}~{}\left({2m\pi kT\over h^{2}}\right)^{3/2}\right]\right\\}$ $None$

Thus entropy defined by (1.20) of a perfect gas is indeed a purely extensive
quantity. Let us also check that (1.20) solve the Gibbs paradox.

Let us consider again the argumentation which leads us to the Gibbs paradox
with the new temperature dependent entropy (just using that $E~{}=~{}3NKT/2$)

$S_{\Lambda}(E,N)~{}=~{}kN\left\\{{5\over 2}+\ln\left[~{}{|\Lambda|\over
N}~{}\left({2m\pi kT\over h^{2}}\right)^{3/2}\right]\right\\}$ $None$

The new variation in entropy is now

$\Delta S~{}=~{}k(N_{A}+N_{B})\Bigg{\\{}{5\over
2}+\ln\Bigg{[}~{}{V_{A}+V_{B}\over N_{A}+N_{B}}~{}\left({2m\pi kT\over
h^{2}}\right)^{3/2}\Bigg{]}\Bigg{\\}}-$ $-kN_{A}\Bigg{\\{}{5\over
2}+\ln\Bigg{[}~{}{V_{A}\over N_{A}}~{}\Big{(}{2m\pi kT\over
h^{2}}\Big{)}^{3/2}\Bigg{]}\Bigg{\\}}-kN_{B}\Bigg{\\{}{5\over
2}+\ln\Bigg{[}~{}{V_{B}\over N_{B}}~{}\Big{(}{2m\pi kT\over
h^{2}}\Big{)}^{3/2}\Bigg{]}\Bigg{\\}}=$ $=kN_{A}\ln\left[{{V_{A}+V_{B}\over
N_{A}+N_{B}}\over{V_{A}\over N_{A}}}\right]+kN_{B}\ln\left[{{V_{A}+V_{B}\over
N_{A}+N_{B}}\over{V_{B}\over N_{B}}}\right]$ $None$

For two different gases this formula gives something similar to (1.17). But if
gas $A$ and gas $B$ are identical, then, since in the initial state and in the
final state temperature and pressure ar unchanged, we must have, by (1.15)

${V_{A}\over N_{A}}~{}=~{}{V_{B}\over N_{B}}~{}=~{}{V_{A}+V_{B}\over
N_{A}+N_{B}}~{}=~{}{kP\over T},~{}~{}~{}~{}{\rm
if~{}gas~{}A~{}and~{}gas~{}B~{}are~{}identical}$ $None$

(of course also for different gases ${V_{A}\over N_{A}}~{}=~{}{V_{B}\over
N_{B}}~{}=~{}{kT\over P}$ but ${V_{A}+V_{B}\over N_{A}+N_{B}}\neq{kT\over
P}$).

Inserting formula (1.23) in (1.22) we obtain

$\Delta S~{}=~{}0~{}~{}~{}~{}~{}~{}~{}~{}{\rm
if~{}gas~{}A~{}and~{}gas~{}B~{}are~{}identical}$

The necessity to divide by the factor $N!$ to escape from the Gibbs paradox is
a new symptom that classical mechanics is not adequate.

####  1.7  The Canonical Ensemble

The Micro Canonical Ensemble is suited for isolated systems where natural
macroscopic variables are the volume $|\Lambda|$, the number of particles $N$
and the energy $E$. We now define a new ensemble which is appropriate to
describe a system which is not isolated, but it is in thermal equilibrium with
a larger system (the heat reservoir), e.g. a gas kept in a box made by heat
conducting walls which is fully immersed in a larger box containing some other
gas at a fixed temperature $T$. Hence this system is constrained to stay in a
box $\Lambda$ with a fixed volume $|\Lambda|$, a fixed number of particles
$N$, at a fixed temperature $T$, but its energy is no longer fixed, since the
system is now allowed to exchange energy with the heat reservoir through the
walls.

We define the Canonical Ensemble for such a system as follows. The space of
configuration of the Canonical Ensemble is

$\Gamma_{c}(\Lambda,N)~{}=~{}\\{(\bm{p},\bm{q}):\bm{p}\in\mathbb{R}^{dN},~{}~{}\bm{q}\in\Lambda^{N}\\}$

The probability measure of the Canonical Ensemble is

$d\mu_{c}(\bm{p},\bm{q})~{}=~{}{1\over Z_{\Lambda}(\beta,N)}{1\over
N!}e^{-\beta H(\bm{p},\bm{q})}{d\bm{p}d\bm{q}\over h^{3N}}$ $None$

where $\beta~{}=~{}(kT)^{-1}$ is a constant proportional to the inverse
temperature of the system ($k$ is again the Boltzmann constant) and the
normalization constant

$Z_{\Lambda}(\beta,N)~{}=~{}{1\over N!}\int e^{-\beta
H(\bm{p},\bm{q})}{d\bm{p}d\bm{q}\over h^{3N}}$ $None$

is the called the partition function of the system in the canonical ensemble.

Thermodynamics is recovered by the following definition. The thermodynamic
function called free energy of the system is obtained in the canonical
ensemble by the formula

$F_{\Lambda}(\beta,N)~{}=~{}-kT\ln Z_{\Lambda}(\beta,N)$ $None$

We now remark that in (1.25) we can perform for free the integration over
momenta.

$\displaystyle Z_{\Lambda}(\beta,N)$ $\displaystyle={1\over
h^{3N}N!}\int\\!\\!dp_{1}\dots\int\\!\\!dp_{N}\int_{\Lambda}\\!\\!dx_{1}\dots\int_{\Lambda}\\!\\!dx_{N}e^{-\beta
H(p_{1},\dots,p_{N},x_{1},\dots,x_{N})}$ $\displaystyle={1\over h^{3N}N!}\int
d{p}_{1}\dots\int
d{p}_{N}\int_{\Lambda}d{x}_{1}\dots\int_{\Lambda}d{x}_{N}e^{-\beta(\sum_{i=1}^{N}{{p}_{i}^{2}\over
2m}+U({x}_{1},\dots x_{N}))}$ $\displaystyle={1\over h^{3N}N!}\int
e^{-\beta{{p}_{1}^{2}\over 2m}}d{p}_{1}\dots\int e^{-\beta{{p}_{N}^{2}\over
2m}}d{p}_{N}\int_{\Lambda}d{x}_{1}\dots\int_{\Lambda}d{x}_{N}e^{-\beta
U({x}_{1},\dots x_{N})}$
$\displaystyle={\left[\int_{-\infty}^{+\infty}e^{-\beta{x^{2}/2m}}dx\right]^{3N}\over
h^{3N}N!}\int_{\Lambda}d{x}_{1}\dots\int_{\Lambda}d{x}_{N}e^{-\beta
U({x}_{1},\dots x_{N})}$ $\displaystyle={\left[(2m\pi/\beta
h^{2})^{3/2}\right]^{N}\over
N!}\int_{\Lambda}d{x}_{1}\dots\int_{\Lambda}d{x}_{N}e^{-\beta U({x}_{1},\dots
x_{N})}$

The integral

${1\over N!}\int_{\Lambda}d{x}_{1}\dots\int_{\Lambda}d{x}_{N}e^{-\beta
U({x}_{1},\dots x_{N})}$

is called the “configurational” partition function of the system. Generally,
it is very difficult to calculate explicitly this function for a real gas. But
in the case of an ideal gas the situation is again immediate. So we can now
give a justification a posteriori for the definition (1.24) by considering
again the case of the ideal gas. Indeed, in the case of the ideal gas we have
that $U({x}_{1},\dots x_{N})=0$ and thus (recall that we have set
$V=|\Lambda|$)

$Z^{\rm ideal\,gas}_{\Lambda}(\beta,N)~{}=~{}{V^{N}\over N!}\left({2m\pi\over
h^{2}\beta}\right)^{3N/2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

Hence, since $\beta^{-1}~{}=~{}kT$ and using also Stirling approximation for
$\ln N!$

$F_{\Lambda}(\beta,N)~{}=~{}-kT\ln
Z_{\Lambda}(\beta,N)~{}=~{}-kTN\left\\{1+\ln\left[{V\over N}\left({2\pi
mkT\over h^{2}}\right)^{3/2}\right]\right\\}$

From free energy we can calculate all thermodynamic quantities. E.g.

$P~{}=~{}-{\partial F\over\partial V}~{}=~{}{NkT\over
V}~{}~{}~{}~{}\Rightarrow~{}~{}~{}~{}PV~{}=~{}NkT$ $S~{}=~{}-{\partial
F\over\partial T}~{}=~{}Nk\left[{5\over 2}+\ln\left[{V\over N}\left({2\pi
mkT\over h^{2}}\right)^{3/2}\right]\right]$ $E~{}=~{}F+TS~{}=~{}{3\over 2}NkT$

Results are identical to the case of Micro Canonical Ensemble!

Note also that the energy $E$ in the canonical ensemble is not fixed and hence
$E$ has to be interpreted as mean energy. This suggests that it could be also
calculated directly by the formula

$E~{}=~{}\langle
H(\bm{p},\bm{q})\rangle~{}=~{}Z^{-1}_{\Lambda}(\beta,N){1\over N!}\int
e^{-\beta H(\bm{p},\bm{q})}H(\bm{p},\bm{q}){d\bm{p}d\bm{q}\over
h^{3N}}=-{\partial\over\partial\beta}\ln Z_{\Lambda}(\beta,N)$

Exercise. Show that $\langle H(\bm{p},\bm{q})\rangle~{}=~{}{3\over 2}NkT$ as
soon as $H(\bm{p},\bm{q})~{}=~{}\sum_{i=1}^{N}{{p}_{i}^{2}\over 2m}$.

####  1.8 Canonical Ensemble: Energy fluctuations

A system in the Canonical Ensemble can have in principle microstates of all
possible energies. This means that the energy fluctuates around its mean value
$E=\langle H(\bm{p},\bm{q})\rangle$. Let us thus check the fluctuations of the
energy in the Canonical Ensemble. The mean energy in the Canonical Ensemble is
given by

$E~{}=~{}\langle H(\bm{p},\bm{q})\rangle~{}=~{}{\int
d\bm{p}d\bm{q}H(\bm{p},\bm{q})e^{-\beta H(\bm{p},\bm{q})}\over\int
d\bm{p}d\bm{q}e^{-\beta H(\bm{p},\bm{q})}}$ $None$

Differentiating both side of (1.27) respect to $\beta$ we get

$\displaystyle{\partial E\over\partial\beta}$ $\displaystyle=-~{}{\int
d\bm{p}d\bm{q}H^{2}(\bm{p},\bm{q})e^{-\beta H(\bm{p},\bm{q})}\int
d\bm{p}d\bm{q}e^{-\beta H(\bm{p},\bm{q})}\over(\int d\bm{p}d\bm{q}e^{-\beta
H(\bm{p},\bm{q})})^{2}}$ $\displaystyle+~{}{\int
d\bm{p}d\bm{q}H(\bm{p},\bm{q})e^{-\beta H(\bm{p},\bm{q})}\int
d\bm{p}d\bm{q}H(\bm{p},\bm{q})e^{-\beta H(\bm{p},\bm{q})}\over(\int
d\bm{p}d\bm{q}e^{-\beta H(\bm{p},\bm{q})})^{2}}$
$\displaystyle~{}=~{}\left[{\int d\bm{p}d\bm{q}H(\bm{p},\bm{q})e^{-\beta
H(\bm{p},\bm{q})}\over\int d\bm{p}d\bm{q}e^{-\beta
H(\bm{p},\bm{q})}}\right]^{2}~{}-~{}{\int
d\bm{p}d\bm{q}H^{2}(\bm{p},\bm{q})e^{-\beta H(\bm{p},\bm{q})}\over\int
d\bm{p}d\bm{q}e^{-\beta H(\bm{p},\bm{q})}}$

Hence we get the relation, for the standard deviation of $E~{}=~{}\langle
H(\bm{p},\bm{q})\rangle$

$\langle H^{2}(\bm{p},\bm{q})\rangle-\langle
H(\bm{p},\bm{q})\rangle^{2}~{}=~{}-{\partial
E\over\partial\beta}~{}=~{}kT^{2}{\partial E\over\partial T}$

From thermodynamics ${\partial E\over\partial T}=C_{V}$ where $C_{V}$ is the
heat capacity. In general $C_{V}\propto N$ as also $E\propto N$ (see e.g. the
case of the perfect gas where $C_{V}={3\over 2}Nk$ and $E={3\over 2}NkT$).
Hence

${\sqrt{\langle H^{2}(\bm{p},\bm{q})\rangle-\langle
H(\bm{p},\bm{q})\rangle^{2}}\over\langle
H(\bm{p},\bm{q})\rangle}~{}=~{}{\sqrt{kT^{2}{\partial E\over\partial T}}\over
E}\approx{1\over\sqrt{N}}\ll 1$

Thus the (relative) fluctuation of the energy around its mean value in the
canonical ensemble are “macroscopically” small (in the sense that they are of
the order of $1/\sqrt{N}$ with $N$ being a very large number). This means that
in the canonical Ensemble it is highly probable to find the system in
microstates with energy equal or very close to the mean energy
$E~{}=~{}\langle H(\bm{p},\bm{q})\rangle$. So Canonical Ensemble is “nearly” a
micro-canonical Ensemble. The energy is not exactly fixed, but it can
fluctuate around a fixed value with relative fluctuations of order
$N^{-1/2}\approx 10^{-12}$.

####  1.9 The Grand Canonical Ensemble

The Micro-Canonical Ensemble applies to isolated systems with fixed $N$, $V$
and $E$, while the Canonical Ensemble describes systems with fixed $N$, $V$
and $T$ and energy variable (e.g. systems in heat bath). The Canonical
Ensemble appears more realistic than the Micro Canonical Ensemble. It is very
difficult to construct a perfectly isolated system, as demanded in the Micro
Canonical Ensemble. So systems whose energy is not known exactly (hence not
perfectly isolated) are easier to construct experimentally.

On the other hand, in the canonical ensemble is still demanded a severe
“microscopic” condition: the number of particles must fixed, i.e. system
confined in $\Lambda$ cannot exchange matter with the outside. This is also a
very difficult situation to create experimentally. We generally deal with
systems where, besides the energy, also the number of particles is not known
exactly. We now thus define the Ensemble suitable to describe systems in
thermodynamic equilibrium in which matter and energy can be exchanged with the
exterior. The fixed thermodynamics parameters for such a system are the volume
$V$, the temperature $T$ and the chemical potential $\mu$.

Hence the configuration space of the Grand canonical Ensemble is

$\Gamma_{GC}(\Lambda)~{}=~{}\bigcup_{N\geq 0}\Gamma_{c}(\Lambda,N)$

(by convention $\Gamma_{0}$ represent the single micro-state in which no
particle is present in the volume $\Lambda$). The (restriction to
$\Gamma_{c}(\Lambda,N)$ of the) probability measure of the Grand Canonical
Ensemble is

$d\mu_{GC}(\bm{p},\bm{q})~{}=~{}{1\over\Xi_{\Lambda}(\beta,\mu)}{e^{N\beta\mu}\over
N!\,h^{3N}}e^{-\beta H(\bm{p},\bm{q})}{d\bm{p}\,d\bm{q}}$ $None$

where again $\beta=1/kT$ is the inverse temperature, $\mu$ is the chemical
potential and the normalization constant

$\Xi_{\Lambda}(\beta,\mu)=\sum_{N=0}^{\infty}{z^{N}\over
h^{3N}N!}\int\\!\\!d{p}_{1}\dots\int\\!\\!d{p}_{N}\int_{\Lambda}\\!\\!d{x}_{1}\dots\int_{\Lambda}\\!\\!d{x}_{N}e^{-\beta
H({p}_{1},\dots,{p}_{N},{x}_{1},\dots,{x}_{N})}$ $None$

is called the partition function in the grand canonical ensemble. Here above
$z~{}\doteq~{}e^{\beta\mu}$ is called ”activity” or ”fugacity” of the system.
The term $N=0$ is put conventionally $=1$ in the sum above while for the term
$N=1$ we have $H({p}_{1},{x}_{1})={{p}_{1}^{2}/2m}$.

Hence $d\mu_{GC}(\bm{p},\bm{q})$ is the probability to find the system in a
micro-state with exactly $N$ particles, with momenta and positions in the
small volume ${d\bm{p}\,d\bm{q}}$ centered at $(\bm{p},\bm{q})$ of the phase
space $\Gamma_{N}$. By convention

$d\mu(\Gamma_{0})~{}=~{}{1\over\Xi(T,\Lambda,\mu)}$

is the probability to find the system in the micro-state where no particle in
$\Lambda$ is present.

Again, since
$H({p}_{1},\dots,{p}_{N},{x}_{1},\dots,{x}_{N})=\sum_{i=1}^{N}\frac{p_{i}^{2}}{2m}+U(x_{1},\dots,x_{N})$,
the integration over momenta can be done explicitly and one gets

$\Xi_{\Lambda}(\beta,\lambda)~{}=~{}\sum_{N=0}^{\infty}{\lambda^{N}\over
N!}\int_{\Lambda}d{x}_{1}\dots\int_{\Lambda}d{x}_{N}~{}e^{-\beta
U({x}_{1},\dots,{x}_{N})}$ $None$

where

$\lambda~{}=~{}e^{\beta\mu}\left({2\pi m\over\beta h^{2}}\right)^{3/2}$ $None$

The parameter $\lambda$ is called configurational activity (or simply activity
when it will be clear from the context).

The connection with thermodynamic in the Grand-canonical ensemble is defined
via the formula

$\beta
P_{\Lambda}(\beta,\lambda)~{}=~{}{1\over|\Lambda|}\ln\Xi_{\Lambda}(\beta,\lambda)$
$None$

and the function $P_{\Lambda}(\beta,\lambda)$ is identified with the
thermodynamical pressure of the system. Another important quantity that can be
calculated from the Grand Canonical partition function is the mean density.
The mean density is obtained by computing, at fixed volume $|\Lambda|$,
temperature $T$ and chemical potential $\mu$ the mean number of particles in
the system.

$\displaystyle\langle N\rangle$
$\displaystyle=~{}{1\over\Xi_{\Lambda}(\beta,\lambda)}\sum_{N=0}^{\infty}{N\lambda^{N}\over
N!}\int_{\Lambda}d{x}_{1}\dots\int_{\Lambda}d{x}_{N}~{}e^{-\beta
U({x}_{1},\dots,{x}_{N})}$ $None$
$\displaystyle=~{}{\lambda}{\partial\over\partial\lambda}{\ln\Xi_{\Lambda}(\beta,\lambda)}$

Hence calling $\rho_{\Lambda}(\beta,\lambda)~{}=~{}{\langle
N\rangle\over|\Lambda|}$

$\rho_{\Lambda}(\beta,\lambda)~{}=~{}{1\over|\Lambda|}{\lambda}{\partial\over\partial\lambda}{\ln\Xi_{\Lambda}(\beta,\lambda)}$
$None$

####  1.10 The ideal gas in the Grand Canonical Ensemble

To conclude this brief introduction let us consider the case of perfect gas in
the Grand Canonical Ensemble. It is very easy to calculate the Grand Canonical
partition function in this case, where $U({x}_{1},\dots,{x}_{N})=0$. E.g., by
(1.30) we get

$\Xi^{ideal\,gas}_{\Lambda}(\beta,\lambda)~{}=~{}\sum_{N=0}^{\infty}{\lambda^{N}\over
N!}~{}\int_{\Lambda}d{x}_{1}\dots\int_{\Lambda}d{x}_{N}~{}=~{}~{}\sum_{N=0}^{\infty}{(\lambda|\Lambda|)^{N}\over
N!}~{}=~{}e^{\lambda|\Lambda|}$

Hence, (1.32) and (1.34) become

$\beta P^{ideal\,gas}~{}=~{}\lambda$ $None$ $\rho^{ideal\,gas}~{}=~{}\lambda$
$None$

In particular (1.36) says that the activity $\lambda$ of a perfect gas
coincides with the its density $\langle N\rangle/|\Lambda|$. Putting (1.36) in
(1.35) we get

$\beta P~{}=~{}\rho$

which is again the equation of state of a perfect gas.

Exercise: Calculate the fluctuation $\langle N^{2}\rangle-\langle
N\rangle^{2}$ in the case of the perfect gas and show that it is of order
$\langle N\rangle$.

####  1.11 The Thermodynamic limit

The dependence of the density $\rho_{\Lambda}(\beta,\lambda)$ from the volume
$\Lambda$ in (1.34) must be a residual one. In fact we expect that, by
increasing the volume $\Lambda$ of our system but keeping fixed the value of
the chemical potential $\mu$ and the inverse temperature $\beta$, the density
of the system does not vary in a sensible way. Values of $|\Lambda|$ that one
can take in thermodynamics are macroscopic, hence very large. We thus may
think that the volume is arbitrarily large (which is the rigorous
formalization of ”macroscopically” large) and define

$\beta
P(\beta,\lambda)~{}=~{}\lim_{\Lambda\to\infty}{1\over|\Lambda|}\ln\Xi_{\Lambda}(\beta,\lambda)$
$None$
$\rho(\beta,\lambda)~{}=~{}\lim_{\Lambda\to\infty}{1\over|\Lambda|}{\lambda}{\partial\over\partial\lambda}{\ln\Xi_{\Lambda}(\beta,\lambda)}$
$None$

The limit $\Lambda\to\infty$ (where the way in which $\Lambda$ goes to
infinity has to be specified in a precise sense) is called the thermodynamic
limit of the Grand Canonical Ensemble. The exact thermodynamic behavior of the
system is recovered at the thermodynamic limit. This limit can be understood
in the physical sense as “the volume macroscopically large”.

Note that, by (1.38) it is possible to express the activity of the system
$\lambda$ as a function of the density $\rho$ and of the (inverse) temperature
$\beta$. So the pressure of the system can be expressed in terms $\rho$ and
$\beta$. This is very easy to do in the case of the ideal gas. When real gases
are concerned (i.e. gases for which the potential energy $U$ is non-zero), the
formula giving the pressure of the system in terms of the density is called
the virial equation of state.

Thermodynamic limit can also be done in the Micro Canonical and Canonical
Ensemble. In this case one has to fix a given density $\rho=N/|\Lambda|$ for
the system and then take the limit $\Lambda\to\infty$, $N\to\infty$ in such
way that $N/|\Lambda|$ is kept constant at the value $\rho$.

In the Micro Canonical Ensemble, in place of (1.5) one define

$S(\rho,E)~{}=~{}\lim_{\Lambda\to\infty,N\to\infty,\atop
N/|\Lambda|,~{}E/|\Lambda|~{}{\rm fixed}}{1\over|\Lambda|}k\ln\left[{1\over
h^{3N}}\Psi_{\Lambda}(E,N)\right]$ $None$

where $S(\rho,E)$ is the infinite-volume specific entropy (i.e. entropy per
unit volume) which is an intensive quantity. In the Canonical Ensemble one can
consider in place of (1.26)

$F(\rho,\beta)~{}=~{}-\lim_{\Lambda\to\infty,N\to\infty,\atop
N/|\Lambda|~{}=~{}\rho}{1\over|\Lambda|}kT\ln Z_{\Lambda}(\beta,N)$ $None$

where $F(\rho,\beta)$ is the infinite volume Gibbs free energy per unit
volume.

In principle the three ensembles that we have considered are equivalent only
at the thermodynamic limit, when the effect of the boundary are removed. Thus
equations of the thermodynamics are exactly recovered at the thermodynamic
limit.

Typical mathematical problems in statistical mechanics are thus to show the
existence of limits (1.40), (1.39) and (1.38) and to show that they produce
the same thermodynamic (otherwise something would be seriously wrong in the
picture of the statistical mechanics).

In the following we will focus our attention mainly on the Grand Canonical
Ensemble and we will investigate the existence of the limit (1.38) and the
property of this limit as a function of $\beta$ and $z$.

###  Chapter 2 The Grand Canonical Ensemble

####  2.1 Conditions on the potential energy

A system of point particles in a volume $\Lambda$ in the Grand Canonical
Ensemble is described by a probability measure on
$\cup_{N}\Gamma_{N}(\Lambda)$ where

$\Gamma_{N}(\Lambda)~{}=~{}\\{({x}_{1},\dots
x_{N})\in\mathbb{R}^{dN}:{x}_{i}\in\Lambda\\}.$

The restriction of this probability measure to $\Gamma_{N}(\Lambda)$ is called
the configurational Gibbs measure (we have already integrated over momenta)

$d\mu({x}_{1},\dots,{x}_{N})~{}=~{}{1\over\Xi_{\Lambda}(\beta,\lambda)}{\lambda^{N}\over
N!}e^{-\beta U({x}_{1},\dots,{x}_{N})}d{x}_{1}\dots d{x}_{N}$ $None$

where $\beta~{}=~{}(kT)^{-1}$ with $T$ absolute temperature and $k$ Boltzmann
constant, while the activity $\lambda$ is given in (1.31).
$d\mu({x}_{1},\dots,{x}_{N})$ is the probability to find the system in the
micro-state in which exactly $N$ particles are present and, for
$i~{}=~{}1,2,\dots,N$, the $i^{th}$ particle is in the small volume
$d^{3}{x}_{i}$ centered at the point ${x}_{i}\in\Lambda$. The normalization
constant $\Xi_{\Lambda}(\beta,\lambda)$ is called the Grand Canonical
partition function of the system and it is given by

$\Xi_{\Lambda}(\beta,\lambda)~{}=~{}1+\sum_{N~{}=~{}1}^{\infty}{\lambda^{N}\over
N!}\int_{\Lambda}d{x}_{1}\dots\int_{\Lambda}d{x}_{N}~{}e^{-\beta
U({x}_{1},\dots,{x}_{N})}$ $None$

The factor $1$ in the sum above corresponds to the micro-state in which no
particle is present, which hence can occur with probability
$\Xi^{-1}(\beta,\Lambda,\lambda)$. The potential energy
$U({x}_{1},\dots,{x}_{N})$ is assumed to be a function
$U:(\mathbb{R}^{d})^{N}\to(\mathbb{R}\cup\\{+\infty\\})$. We will suppose from
now on that $U({x}_{1},\dots,{x}_{N})$ has the following form

$U({x}_{1},\dots,{x}_{N})~{}=~{}\sum_{1\leq i<j\leq
N}V({x}_{i}-{x}_{j})+\sum_{i=1}^{N}\Phi_{\rm e}({x}_{i})$

where $V({x})$ is a function $V:\mathbb{R}^{d}\to\mathbb{R}\cup\\{+\infty\\}$
and $\Phi_{\rm e}({x})$ is a function $\Phi_{\rm
e}:\mathbb{R}^{d}\to\mathbb{R}$. We will always assume that $V({x})$ is such
that $V(-{x})~{}=~{}V({x})$. We let $|x|$ to denote the Euclidean norm of $x$.
Physically the assumption above on the potential energy means that we are
restricting to the case of particles interacting via a translational invariant
pair potential $V$ plus an external potential $\Phi_{\rm e}$. The interaction
$\sum_{1\leq i<j\leq N}V({x}_{i}-{x}_{j})$ is “internal” in the sense that it
depends only on mutual positions in space of particles (i.e. only from vectors
${x}_{i}-{x}_{j}$). The interaction $\sum_{i=1}^{N}\Phi_{\rm e}({x}_{i})$
depends instead on the absolute positions of particles in space and $\Phi_{\rm
e}({x}_{i})$ is interpreted as the effect of the world “outside” the boundary
of $\Lambda$ on the $i^{th}$ particle confined in $\Lambda$. For instance,
suppose that outside $\Lambda$ there are $M$ particles in fixed positions
$y_{1},\dots y_{M}$, then $\Phi_{\rm
e}({x}_{i})~{}=~{}\sum_{j~{}=~{}1}^{M}V({x}_{i}-y_{j})$.

The choice of $\Phi_{\rm e}$ is somehow arbitrary, in the sense that it will
depend on conditions we are supposing outside $\Lambda$. A given choice of
$\Phi_{\rm e}$ is called generally a boundary condition. For continuous
systems the question of the effect of the boundary conditions at the
thermodynamic limit is rather difficult. The pressure is expected to be
independent on the choice of the boundary condition but other quantities (such
as some derivatives of the pressure) may not be independent on boundary
conditions. In this section we will make the mathematically simplest choice
$\Phi_{\rm e}=0$ (i.e. no influence at all on particles inside $\Lambda$ from
the world outside) which is called free boundary condition. We will consider
in this chapter just free boundary conditions, hence we will suppose

$U({x}_{1},\dots,{x}_{N})~{}=~{}\sum_{1\leq i<j\leq N}V({x}_{i}-{x}_{j})$
$None$

By (2.3), interaction energy between particles is known once we have specified
the two body potential $V({x})$. We immediately see that the function
$U({x}_{1},\dots,{x}_{N})$ defined in (2.3) has the following properties. $i)$
$U$ is symmetric for the exchange of particles.

Let $\\{\sigma(1),\sigma(2),\dots,\sigma(N)\\}$ be a permutation of the set
$\\{1,2,\dots,N\\}$ then

$U({x}_{\sigma(1)},\dots,{x}_{\sigma(N)})~{}=~{}U({x}_{1},\dots,{x}_{N})$
$None$

$ii)$ $U$ is translational invariant.

Namely, if $(x_{1},\dots,x_{N})$ and $(x^{\prime}_{1},\dots,x^{\prime}_{N})$
are two configurations which differs only by a translation then

$U({x}_{1},\dots,{x}_{N})~{}=~{}U({x}^{\prime}_{1},\dots,{x}^{\prime}_{N})$
$None$

Some further conditions on the potential $V$ must be imposed. Stability and
temperedness are commonly considered as minimal conditions to guarantee a good
statistical mechanics behavior of the system (see, e.g., [44] and [15]).

###### Definition 2.1

A pair potential $V(x)$ is said to be stable if there exists $C\geq 0$ such
that, for all ${n}\in\mathbb{N}$ such that $n\geq 2$ and all
$(x_{1},\dots,x_{n})\in\mathbb{R}^{dn}$,

$\sum_{1\leq i<j\leq n}V(x_{i}-x_{j})\geq-nC$ $None$

###### Definition 2.2

A pair potential $V(x)$ is said to be regular

$\int_{\mathbb{R}^{d}}|e^{-\beta V(x)}-1|dx<\infty$ $None$

A typical example of a regular pair potential $V(x)$ is a potential which is
absolutely integrable when $|x|$ is outside a sphere of radius $r_{0}>0$ and
it is positive for $|x|\leq r_{0}$. Then for such a potential, letting
$S_{d}(r_{0})$ be the volume of the $d$-dimensional sphere of radius $r_{0}$
and letting $C=\int_{|x|>r_{0}}|V(x)|dx$,

$\displaystyle\int_{\mathbb{R}^{d}}|e^{-\beta V(x)}-1|dx$
$\displaystyle=\int_{|x|\leq r_{0}}|e^{-\beta
V(x)}-1|dx+\int_{|x|>r_{0}}|e^{-\beta V(x)}-1|dx$ $\displaystyle\leq
S_{d}(r_{0})+\int_{|x|>r_{0}}\left[\sum_{n=1}^{\infty}{|-\beta V(x)|^{n}\over
n!}\right]dx$ $\displaystyle\leq
S_{d}(r_{0})+\sum_{n=1}^{\infty}{\beta^{n}\over
n!}\int_{|x|>r_{0}}|V(x)|^{n}dx$ $\displaystyle\leq
S_{d}(r_{0})+\sum_{n=1}^{\infty}{\beta^{n}\over
n!}\left(\int_{|x|>r_{0}}|V(x)|dx\right)^{n}$ $\displaystyle\leq
S_{d}(r_{0})+\sum_{n=1}^{\infty}{\beta^{n}\over n!}C^{n}$
$\displaystyle=S_{d}(r_{0})+e^{\beta C}-1$ $\displaystyle<+\infty$

A stable and regular pair potential $V(x)$ will be also called admissible.

###### Definition 2.3

Given $V(x)$ admissible, and given $n\geq 2$ let

$B_{n}=\sup_{(x_{1},\dots,x_{n})\in\mathbb{R}^{dn}}-{1\over n}\sum_{1\leq
i<j\leq n}V(|x_{i}-x_{j}|)$ $None$

then the nonnegative number

$B=\sup_{n\geq 2}B_{n}$ $None$

is called the stability constant of the potential $V(x)$.

Note that $B$ is non-negative and $B=0$ if and only if $V(|x|)\geq 0$ (i.e.
“repulsive” potential). Stability and regularity are actually deeply
interconnected and the lack of one of them always produces non thermodynamic
or catastrophic behaviors (see ahead).

The conditions $i)$ and $ii)$ are motivated by physical considerations. They
originate from the observation that most of the physical interactions are
indeed symmetric under exchange of particles and translational invariant.

Stability and regularity are from the physical point of view more difficult to
understand. Concerning in particular regularity, it is quite natural to assume
that potential must vanish at large distances since particles far away are
expected to interact in a negligible way. On the other hand, it is not clear
why the rate of decay has to be such that (2.7) is satisfied. Concerning the
stability condition, we will see that it is somehow related to the fact that
particles are not allowed to be (or pay a high price to be) at short distance
from each other.

We will see below that the grand-canonical partition defined in (1.1) is a
holomorphic function of $\lambda$ if the potential $V(|x|)$ is stable.
Moreover, under very mild additional conditions on the potential (upper-
continuity) it can be proved that the converse is also true (see [44]). In
other words $\Xi_{\Lambda}(\beta,\lambda)$ converges if and only if the
potential $V$ is stable. So, in some sense, stability is a sine qua non
condition to construct a consistent statistical mechanics for continuous
particle systems.

###### Proposition 2.1

Let $U({x}_{1},\dots,{x}_{N})$ be a stable interaction with stability constant
$B$ and let $\lambda$ in (2.2) be allowed to vary in $\mathbb{C}$, then series
in the r.h.s. of (2.2) converges absolutely for all $\lambda\in\mathbb{C}$,
all $\beta\in\mathbb{R}^{+}$ and all $\Lambda$ Lebesgue measurable set in
$\mathbb{R}^{d}$, or in other words the function $\Xi(\beta,\Lambda,\lambda)$,
as a function of $\lambda$ in the complex plane, is holomorphic.

Proof.

$\displaystyle|\Xi_{\Lambda}(\beta,\lambda)|$ $\displaystyle\leq
1+\sum_{N=1}^{\infty}{|\lambda|^{N}\over
N!}\int_{\Lambda}d{x}_{1}\dots\int_{\Lambda}d{x}_{N}~{}e^{-\beta
U({x}_{1},\dots,{x}_{N})}$
$\displaystyle\leq~{}1+\sum_{N=1}^{\infty}{|\lambda|^{N}\over
N!}\int_{\Lambda}d{x}_{1}\dots\int_{\Lambda}d{x}_{N}e^{+\beta BN}$
$\displaystyle=~{}\sum_{N=0}^{\infty}{(|\Lambda||\lambda|e^{\beta B})^{N}\over
N!}$ $\displaystyle=~{}\exp\\{|\Lambda||\lambda|e^{\beta B}\\}$

$\Box$

The stability condition is therefore a sufficient condition for the absolute
convergence of the Grand Canonical partition function. As mentioned above, it
can be also shown, that the converse also holds, i.e. stability condition is
also a necessary condition (see [44]). But in that case some further
conditions on the function $V$ are needed. We rather prefer here to show by
mean of examples how violation of stability and/or temperness produce non
thermodynamic behaviors in the system. In all examples to follow we will
assume to work in $\mathbb{R}^{3}$.

####  2.2 Potentials too attractive at short distances

A simple way to violate stability is by choosing a potential $V({x})$ which is
negative in the neighborhood of ${x}=0$. For example, let us consider the
potential $V_{1}^{\rm bad}(|{x}|)$ as in figure 1: a continuous, not
decreasing, compactly supported function of $|x|$. This potential is
continuous, tempered (i.e. satisfies (2.7)), bounded (i.e. $|V_{1}^{\rm
bad}(|{x}|)|\leq\alpha$, for some $\alpha>0$), and strictly negative around
the origin ${x}~{}=~{}0$ (i.e. $\exists\delta>0$ and $b>0$ such that
$V_{1}^{\rm bad}(|{x}|)\leq-b$ whenever $|{x}|\leq~{}\delta$). This potential
$V_{1}^{\rm bad}(|{x}|)$ is not stable. Indeed, if we place $N$ particles in
positions ${x}_{1},\dots,{x}_{N}$ so close one to each other that
$|{x}_{i}-{x}_{j}|\leq\,\delta$ (for all $i,j~{}=~{}1,2,\dots,N$), then
$U({x}_{1},\dots,{x}_{N})\leq-b{N(N-1)/2}$.

Figure 1. The potential $V^{\rm bad}_{1}(|{x}|)$

As we said above, it can be shown that the lack of stability destroys the
convergence of the series for $\Xi_{\Lambda}(\beta,\lambda)$. I.e. it is
possible to show that the grand canonical partition function with the
potential $V_{1}^{\rm bad}(|{x}|)$ is a divergent series. Even so, one may
argue that the partition function in the canonical ensemble for the potential
$V_{1}^{\rm bad}(|{x}|)$ is still well defined and calculations could then be
performed in this ensemble.

Let us therefore do these calculations in the (configurational) Canonical
Ensemble with $\beta$ and $N$ fixed.

First we consider a catastrophic situation in which the $N$ particles collapse
in a small region inside $\Lambda$. Namely, we calculate the probability to
find the system in a micro-state with the $N$ particles being all located in a
small sphere $S_{\delta}\subset\Lambda$ of radius $\delta/2$ (so that they are
all at distance less or equal to $\delta$). By (2.1) a lower bound for such
probability is given by

$\displaystyle P_{\rm bad}(N)$ $\displaystyle=~{}{1\over
Z_{\Lambda}(\beta,N)}\int_{S_{\delta}}d{x}_{1}\dots\int_{S_{\delta}}d{x}_{N}{1\over
N!}e^{-\beta U({x}_{1},\dots,{x}_{N})}$ $\displaystyle\geq~{}{1\over
Z_{\Lambda}(\beta,N)}\left[{\pi\delta^{3}\over 6}\right]^{N}{1\over
N!}e^{+\beta b{N(N-1)\over 2}}$

Now consider configurations “macroscopically correct”, i.e. micro-states with
$N$ particles in positions ${x}_{1},\dots x_{N}$ uniformly distributed in the
box $\Lambda$, with density equal to the density $\rho~{}=~{}{N/|\Lambda|}$
fixed by the parameters $N$ and $|\Lambda|$ in the canonical ensemble. If the
potential $V_{1}^{\rm bad}$ is tempered then it is not difficult to see that
for such configurations $|U({x}_{1},\dots,{x}_{N})|\leq CN\rho$ where
$2C~{}=~{}\int_{\mathbb{R}^{3}}d{x}|V({x})|$. As a matter of fact, let us
consider the box $\Lambda$ as the disjoint union of small cubes $\Delta$ (with
volume $|\Delta|$) so that each of the particles $x_{1}\dots,x_{N}$ belongs to
one of these small cubes $\Delta$ (with volume $|\Delta|$). Hence, given a
configuration $x_{1},\dots,x_{N}$ (recall that $V_{1}^{\rm bad}({x})$ non-
positive by assumption)

$\displaystyle\sum_{j\in\\{1,2,\dots,N\\}\atop j\neq i}V_{1}^{\rm
bad}(|{x}_{j}-{x}_{i}|)$
$\displaystyle=~{}\sum_{\Delta\subset\Lambda}\sum_{j\in\\{1,2,\dots,N\\}\atop\;{x}_{j}\in\Delta,~{}j\neq
i}V_{1}^{\rm bad}(|{x}_{j}-{x}_{i}|)|$
$\displaystyle\geq\sum_{\Delta\subset\Lambda}V_{1}^{\rm
bad}(r_{\Delta})\sum_{j\in\\{1,2,\dots,N\\}\atop\;{x}_{j}\in\Delta,\;j\neq
i}1$

where in the last line to get the inequality we have used the assumption that
$V_{1}^{\rm bad}$ is non decreasing and $\sum_{\Delta\subset\Lambda}$ runs
over all small cubes whose disjoint union is $\Lambda$ and $r_{\Delta}$
denotes the (minimal) distance of a point inside the cube $\Delta$ from the
point ${x}_{i}$. Now, since we are assuming that particles are uniformly
distributed in $\Lambda$ and choosing the dimensions of the small cubes
sufficiently large in order to still consider these cubes macroscopic, so that
the particles in a small cube $\Delta$ are still uniformly distributed with
density $\rho$ or very close to $\rho$. Hence we may assume that there exists
$\varepsilon>0$ such that, for each $\Delta\subset\Lambda$

$\sum_{j\in\\{1,2,\dots,N\\}\atop\;{x}_{j}\in\Delta\;j\neq
i}1~{}\leq(1+\varepsilon)\rho|\Delta|$

so that

$\displaystyle\sum_{j\in\\{1,2,\dots,N\\}:\;j\neq i}V_{1}^{\rm
bad}(|{x}_{j}-{x}_{i}|)$
$\displaystyle\geq\rho(1+\varepsilon)\,\mbox{$\sum_{\Delta}$}V_{1}^{\rm
bad}(r_{\Delta})\Delta$ $None$
$\displaystyle\geq\rho(1+\varepsilon)\int_{\Lambda}V_{1}^{\rm
bad}(|{x}-{x}_{i}|)d{x}$
$\displaystyle\geq\rho(1+\varepsilon)\int_{\mathbb{R}^{3}}V_{1}^{\rm
bad}({x})d{x}$

Hence, for such configurations we have

$\displaystyle\sum_{1\leq i<j\leq N}V_{1}^{\rm bad}({x}_{i}-{x}_{j})$
$\displaystyle=~{}{1\over 2}\sum_{i=1}^{N}\sum_{j:~{}j\neq i}V_{1}^{\rm
bad}({x}_{i}-{x}_{j})$ $\displaystyle\geq~{}{N\rho(1+\varepsilon)\over
2}\int_{\mathbb{R}^{3}}V_{1}^{\rm bad}({x})d{x}$ $\displaystyle=~{}-NC\rho$

with

$C={(1+\varepsilon)\over 2}\left[-\int_{\mathbb{R}^{3}}V_{1}^{\rm
bad}({x})dx\right]$

positive (again, recall that $V_{1}^{\rm bad}({x})$ non-positive by
assumption). Then an upper bound for the probability for such configuration to
occur is, according with (2.1)

$\displaystyle P_{\rm good}(N)$ $\displaystyle=~{}{1\over
Z_{\Lambda}(\beta,N)}\int_{\Lambda}d{x}_{1}\dots\int_{\Lambda}d{x}_{N}{1\over
N!}e^{-\beta U({x}_{1},\dots,{x}_{N})}$ $\displaystyle\leq~{}{1\over
Z_{\Lambda}(\beta,N)}|\Lambda|^{N}{1\over N!}e^{+\beta\rho CN}$
$\displaystyle=~{}{1\over Z_{\Lambda}(\beta,N)}N^{N}\rho^{-N}{1\over
N!}e^{+\beta C\rho N}$

Hence a lower bound for the ratio between the probability of bad
configurations and good configurations is

${P_{\rm bad}(N)\over P_{\rm good}(N)}~{}\geq~{}{\left[{\pi\over
6}\rho\delta^{3}\right]^{N}{1\over N!}e^{+\beta b{N(N-1)\over 2}}\over
N^{N}{1\over N!}e^{+\beta C\rho N}}~{}=~{}\left[{\pi\over
6}\rho\delta^{3}\over e^{\beta({b\over 2}+C\rho)}\right]^{N}{e^{+\beta{b\over
2}{N^{2}}}\over N^{N}}$

and, no matter how small $\delta$ and/or $b$ are we have that

$\lim_{N\to\infty}{P_{\rm bad}(N)\over P_{\rm good}(N)}~{}=~{}+\infty$

This means that it is far more probable to find the system in a micro-state in
which all particles are all contained within a small sphere of diameter
$\delta$ in some place of $\Lambda$ rather than in a micro-state
“macroscopically correct”, i.e. a configuration with particles uniformly
distributed in $\Lambda$ with a constant density $\rho=N/|\Lambda|$.

####  2.3  Two examples of infrared catastrophe

In the previous section we consider a non-stable (but regular) pair potential
$V_{1}^{\rm bad}(|{x}|)$ which was not preventing particles to accumulate in
arbitrarily small regions of the space. We now show that the lack of stability
and/or regularity of a pair potential, even preventing particles to accumulate
in arbitrarily small regions of the space, yields to non-thermodynamic
situations.

#####  2.3.1 Potential with hard core too attractive at large distances

We consider a second case of “bad” potential which this time does not allow
particles to come together at arbitrarily short distances but it is too
attractive at large distances. Let the space dimension be set at $d=3$ and let
$a>0$, $3>\varepsilon>0$ and define,

$V_{2}^{\rm bad}(|{x}|)~{}=~{}\begin{cases}+\infty&{\rm if}~{}|{x}|\leq a\\\
{-|{x}|^{-3+\varepsilon}}&{\rm otherwise}\end{cases}$ $None$

Figure 2. The potential $V^{\rm bad}_{2}(|{x}|)$

It can be proved that this potential, illustrated in figure 2, is neither
stable nor tempered. This is a first example of a so called hard-core type
potential (where the hard core condition is $V_{2}^{\rm bad}(x)=+\infty$ if
$|{x}|\leq a$). It describes a system of interacting hard spheres of radius
$a$. In fact, since $V_{2}^{\rm bad}(|{x}|)$ is $+\infty$ whenever $|{x}|\leq
a$, then $U({x}_{1},\dots,{x}_{N})=+\infty$ whenever $|{x}_{i}-{x}_{j}|\leq a$
for some $i,j$, thus the Gibbs factor for such configuration (i.e where some
particles are at distances less or equal $a$) is $\exp\\{-\beta
U({x}_{1},\dots,{x}_{N})\\}~{}=~{}0$ and hence it has zero probability to
occur.

This means that such system cannot take densities greater than a certain
density $\rho_{\rm cp}=c/a^{3}$ called the close-packing density (here $c$ is
a suitable constant111The constant for the $d=3$ case is actually known:
$\pi/2\sqrt{3}$.), where particles are as near as possible one to each other
compatibly with the hard core condition. E.g., particles are arranged (in a
close-packed way) at vertices of a face-centered cubic lattice with nearest
neighbors at distance $a$.

Let us again consider the system in the Canonical Ensemble at fixed inverse
temperature $\beta$, fixed volume $|\Lambda|$ and fixed number of particles
$N$, hence at fixed $\rho~{}=~{}N/V$. We consider the case $d=3$ and choose
$N$ and $|\Lambda|$ in such way that $\rho$ is much smaller that the close-
packing density $\rho_{\rm cp}$, i.e. $\rho/\rho_{\rm cp}\ll 1$.

We now compare the probability to find the system in a region of the space-
phase constituted by micro-states near the close-packing situation. To be
specific, let $y_{1},\dots y_{N}$ be sites of a face-centered cubic lattice
with nearest neighbors at distance $a+\delta$ (with $\delta$ much smaller than
$a$) organized in a close-packed configuration in such way that they fill
completely a cube $\Lambda_{\rm cp}$ of size $L_{\rm cp}$. We now suppose that
the $N$ particles are at positions $x_{1},\dots x_{N}$ such that
$|x_{i}-y_{i}|<{\delta\over 2}$ for $i=1,\dots N$. In other words the $i^{\rm
th}$ particle can move freely inside the sphere $S_{\delta}^{i}$ of radius
$\delta/2$ centered at the vertex $y_{i}$ of the fcc lattice without violating
the hard core condition of the pair potential. In this set of configurations
the density can vary from the maximum $\rho_{\rm cp}={c\over a^{3}}$ to a
minimum density which is surely greater than $\tilde{\rho}_{\rm
cp}={c\over(a+2\delta)^{3}}=(1+{2\delta\over a})^{-3}\rho_{\rm cp}$. In such
configurations the system does not occupy uniformly all the available volume
$|\Lambda|$, but it rather occupies the fraction ${\rho\over\rho_{\rm cp}}$ of
the available volume $\Lambda$ (since $N=\rho_{\rm cp}|\Lambda_{\rm
cp}|=\rho|\Lambda|$ and thus $|\Lambda_{\rm
cp}|~{}=~{}|\Lambda|{\rho\over\rho_{\rm cp}}$) leaving a region (with volume
$|\Lambda|{(\rho_{\rm cp}-\rho)/\rho_{\rm cp}}$) empty inside $\Lambda$. Of
course such configurations are non thermodynamics. To fix the ideas and
simplify the calculations let us suppose that $\Lambda$ is a sphere of radius
$R$ while we recall that by assumption $\Lambda_{\rm cp}$ is a cube of size
$L_{\rm cp}$. Let of course further assume that $a$ is much smaller than
$L_{\rm cp}$ (actually $a<{L_{\rm cp}\over 2^{1/\varepsilon}}$ will do).

In these configurations the potential energy $U$ is strongly negative. An
upper bound for the value of $U$ for such type of configurations can be
obtained by a reasoning similar to that took us to the (2.10).

Indeed, suppose that particles are arranged in a configuration near the close
packing (in the sense specified above). Then as before

$\displaystyle\sum_{j\in\\{1,2,\dots,N\\}\atop j\neq i}V_{2}^{\rm
bad}(|{x}_{j}-{x}_{i}|)$
$\displaystyle=~{}\sum_{\Delta\subset\Lambda}\sum_{j\in\\{1,2,\dots,N\\}\atop\;j\neq
i,\,{x}_{j}\in\Delta}V_{2}^{\rm bad}(|{x}_{j}-{x}_{i}|)|$
$\displaystyle\leq~{}\sum_{\Delta\subset\Lambda}V_{2}^{\rm bad}(r^{\rm
max}_{\Delta})\sum_{j\in\\{1,2,\dots,n\\}\atop j\neq i,\;{x}_{j}\in\Delta}1$

where this time $r^{\rm max}_{\Delta}$ represent the maximal distance of a
point in the small cube $\Delta$ from $x_{i}$. Now we have that

$\sum_{j\in\\{1,2,\dots,n\\}\atop j\neq
i,\;{x}_{j}\in\Delta}1~{}\geq\tilde{\rho}_{\rm cp}|\Delta|$

Moreover there exists surely an $\eta_{\Delta}$ (depending on $\Delta$) such
that

$\sum_{\Delta\subset\Lambda}V_{2}^{\rm bad}(r^{\rm
max}_{\Delta})|\Delta|~{}\leq~{}-(1-\eta_{\Delta})B_{\varepsilon}(\Lambda_{\rm
cp},x_{i})$

where $\Lambda_{\rm cp}$ denotes the region inside $\Lambda$ where the $N$
close-packed particles are situated (which, recall, is assumed to be a cube
contained in the sphere $\Lambda$) and

$B_{\varepsilon}(\Lambda_{\rm cp},x_{i})~{}=~{}\int_{{x}\in\Lambda_{\rm
cp},~{}|{x}-x_{i}|>a}{1\over|{x}-x_{i}|^{3-\varepsilon}}d{x}$

Now, since we are assuming that $\Lambda_{\rm cp}$ is a square then, wherever
$x_{i}$ is situated inside $\Lambda_{\rm cp}$, there is always a quarter of
sphere, say $U$, of radius $L_{\rm cp}/2$ all contained in $\Lambda_{\rm cp}$
such that $x_{i}$ is the center of this sphere and hence

$\displaystyle B_{\varepsilon}(\Lambda_{\rm cp},x_{i})$
$\displaystyle=~{}\int_{{x}\in\Lambda_{\rm
cp},~{}|{x}-x_{i}|>a}{1\over|{x}-x_{i}|^{3-\varepsilon}}d{x}$
$\displaystyle\geq~{}\int_{x\in
U,~{}|{x}|>a}{1\over|{x}|^{3-\varepsilon}}d{x}$
$\displaystyle=~{}\int_{0}^{\pi\over 2}d\varphi\int_{0}^{\pi\over 2}\sin\theta
d\theta\int_{a}^{L_{\rm cp}\over 2}{r^{2}\over r^{3-\varepsilon}}dr$
$\displaystyle=~{}{\pi\over 2\varepsilon}(L_{\rm
cp}^{\varepsilon}-a^{\varepsilon})$ $\displaystyle\geq~{}{\pi\over
4\varepsilon}L_{\rm cp}^{\varepsilon}$

Therefore we get, observing that $L^{\varepsilon}_{\rm cp}=\Lambda_{\rm
cp}^{\varepsilon\over 3}$

$\sum_{j\in\\{1,2,\dots,N\\}\atop j\neq i}V_{2}^{\rm
bad}(|{x}_{j}-{x}_{i}|)~{}\leq~{}-(1-\eta_{\Delta})\tilde{\rho}_{\rm
cp}{\pi\over 4\varepsilon}|\Lambda_{\rm cp}|^{\varepsilon\over 3}$

and therefore

$\displaystyle U({x}_{1},\dots,{x}_{N})$ $\displaystyle=~{}{1\over
2}\sum_{i=1}^{N}\sum_{j\in\\{1,2,\dots,N\\}\atop j\neq i}V_{2}^{\rm
bad}(|{x}_{j}-{x}_{i}|)$ $\displaystyle\leq~{}-{N}C\rho_{\rm cp}|\Lambda_{\rm
cp}|^{\varepsilon\over 3}$

with

$C={1\over 2}(1+{2\delta\over a})^{-3}(1-\eta_{\Delta}){\pi\over
4\varepsilon}$ $None$

This allows to bound from below the probability $P_{\rm bad}(N)$ to find the
system (in the canonical ensemble) in such bad configurations near the close-
packing as follows.

$\displaystyle P_{\rm bad}(N)$ $\displaystyle=~{}{1\over
Z_{\Lambda}(\beta,N)}\int_{S^{1}_{\delta}}d{x}_{1}\dots\int_{S^{N}_{\delta}}d{x}_{N}{1\over
N!}e^{-\beta U({x}_{1},\dots,{x}_{N})}$ $\displaystyle\geq{1\over
Z_{\Lambda}(\beta,N)}\left[{4\over 3}\pi\delta^{3}\right]^{N}{1\over
N!}e^{+\beta{N}C\rho_{\rm cp}|\Lambda_{\rm cp}|^{\varepsilon\over 3}}$

As far as good configurations (i.e. those with uniform density
$\rho=N/|\Lambda|$) are concerned, proceeding similarly one can bound

$\displaystyle\sum_{j\in\\{1,2,\dots,N\\}\atop j\neq i}V_{2}^{\rm
bad}(|{x}_{j}-{x}_{i}|)$
$\displaystyle=~{}\sum_{\Delta\subset\Lambda}\sum_{j\in\\{1,2,\dots,N\\}\atop\;{x}_{j}\in\Delta,~{}j\neq
i}V_{2}^{\rm bad}(|{x}_{j}-{x}_{i}|)|$
$\displaystyle\geq\sum_{\Delta\subset\Lambda}V_{2}^{\rm bad}(r^{\rm
min}_{\Delta})\sum_{j\in\\{1,2,\dots,N\\}\atop\;{x}_{j}\in\Delta,\;j\neq i}1$

where this time $r^{\rm min}_{\Delta}$ represent the minimal distance of a
point in the small cube $\Delta$ from $x_{i}$. Since, as before, we are
assuming that particles are uniformly distributed in $\Lambda$ with density
$\rho$ we may assume that there exists a small $\gamma>0$ such that, for each
$\Delta\subset\Lambda$

$\sum_{j\in\\{1,2,\dots,N\\}\atop\;{x}_{j}\in\Delta\;j\neq
i}1~{}\leq(1+\gamma)\rho|\Delta|$

so that

$\sum_{j\in\\{1,2,\dots,N\\}\atop j\neq i}V_{2}^{\rm
bad}(|{x}_{j}-{x}_{i}|)~{}\geq~{}\rho(1+\gamma)\,\sum_{\Delta\subset\Lambda}V_{1}^{\rm
bad}(r^{\rm min}_{\Delta})|\Delta|$

Now we have

$\sum_{\Delta\subset\Lambda}V_{1}^{\rm bad}(r^{\rm min}_{\Delta})|\Delta|\geq-
B_{\varepsilon}(\Lambda,x_{i})$

with (recall that $\Lambda$ is a sphere of radius $R$ and thus
$R=(3/4\pi)^{1\over 3}|\Lambda|^{1\over 3}$)

$\displaystyle B_{\varepsilon}(\Lambda,x_{i})$
$\displaystyle=~{}\int_{{x}\in\Lambda,~{}|{x}-x_{i}|>a}{1\over|{x}-x_{i}|^{3-\varepsilon}}d{x}$
$\displaystyle\leq~{}\int_{{x}\in\Lambda,~{}|{x}|>a}{1\over|{x}|^{3-\varepsilon}}d{x}$
$\displaystyle={4\pi\over\varepsilon}(R^{\varepsilon}-a^{\varepsilon})$
$\displaystyle\leq{3^{\varepsilon\over 3}(4\pi)^{1+{\varepsilon\over
3}}\over\varepsilon}|\Lambda|^{\varepsilon\over 3}$

Hence

$\sum_{j\in\\{1,2,\dots,N\\}\atop j\neq i}V_{2}^{\rm
bad}(|{x}_{j}-{x}_{i}|)~{}\geq~{}-\rho(1+\gamma)\,{3(4\pi)^{1+{\varepsilon\over
3}}\over\varepsilon}|\Lambda|^{\varepsilon\over 3}$

and therefore for the “good” configurations $x_{1},\dots,x_{N}$ we have

$\displaystyle U(x_{1},\dots,x_{N})$ $\displaystyle=~{}\sum_{1\leq i<j\leq
N}V_{1}^{\rm bad}({x}_{i}-{x}_{j})$ $\displaystyle=~{}{1\over
2}\sum_{i=1}^{N}\sum_{j:~{}j\neq i}V_{1}^{\rm bad}({x}_{i}-{x}_{j})$
$\displaystyle\geq~{}-N~{}{(1+\gamma)\over 2}{3(4\pi)^{1+{\varepsilon\over
3}}\over\varepsilon}\rho|\Lambda|^{\varepsilon\over 3}$
$\displaystyle=~{}-NC^{\prime}\rho|\Lambda|^{\varepsilon\over 3}$

with

$C^{\prime}={(1+\gamma)\over 2}{3(4\pi)^{1+{\varepsilon\over
3}}\over\varepsilon}$ $None$

Thus an upper bound for the “good” configurations is

$\displaystyle P_{\rm good}(N)$ $\displaystyle=~{}{1\over
Z_{\Lambda}(\beta,N)}\int_{\Lambda}d{x}_{1}\dots\int_{\Lambda}d{x}_{N}{1\over
N!}e^{-\beta U({x}_{1},\dots,{x}_{N})}$ $\displaystyle\leq~{}{1\over
Z_{\Lambda}(\beta,N)}|\Lambda|^{N}{1\over
N!}e^{+\beta{N}C^{\prime}\rho|\Lambda|^{\varepsilon\over 3}}$

Hence the ratio between the probability of bad and good configurations is

${P_{\rm bad}(N)\over P_{\rm good}(N)}\geq{\left[{4\over
3}\pi\delta^{3}\right]^{N}e^{+\beta NC\rho_{\rm cp}|\Lambda_{\rm
cp}|^{\varepsilon\over 3}}\over|\Lambda|^{N}e^{+\beta
NC^{\prime}\rho|\Lambda|^{\varepsilon\over 3}}}$

recalling that

$|\Lambda|~{}=~{}N/\rho~{}~{}~{}~{}~{}~{}|\Lambda_{\rm cp}|~{}=~{}N/\rho_{\rm
cp}~{}$

we get

${P_{\rm bad}(N)\over P_{\rm good}(N)}\geq\left[{4\over
3}\pi\delta^{3}\rho\right]^{N}{e^{+\beta CN^{1+{\varepsilon\over
3}}\left[\rho^{1-{\varepsilon\over 3}}_{\rm cp}-{C^{\prime}\over
C}\rho^{1-{\varepsilon\over 3}}\right]}\over N^{N}}$

Now observe that factor $\left[\rho^{1-{\varepsilon\over 3}}_{\rm
cp}-{C^{\prime}\over C}\rho^{1-{\varepsilon\over 3}}\right]$ in the
exponential is positive if $\rho$ is sufficiently smaller than $\rho_{\rm
cp}$, i.e. , recalling (2.12) and (2.13), if

${\rho\over\rho_{\rm cp}}<{1-\eta_{\Delta}\over 1+\gamma}{1\over
16(12\pi)^{\varepsilon\over 3}(1+{2\delta\over a})^{3}}$

Hence, calling

$C_{1}~{}=~{}e^{+\beta C\left[\rho^{1-{\varepsilon\over 3}}_{\rm
cp}-{C^{\prime}\over C}\rho^{1-{\varepsilon\over
3}}\right]},~{}~{}~{}~{}C_{2}~{}=~{}\left[{4\over
3}\pi\delta^{3}\rho\right]^{-1}$

and noting that $C_{1}>1$ (for $\rho$ sufficiently smaller than $\rho_{\rm
cp}$) we get

${P_{\rm bad}(N)\over P_{\rm good}(N)}\geq{C_{1}^{N^{1+{\varepsilon\over
3}}}\over C_{2}^{N}N^{N}}$

It is just a simple exercise to show that, if $C_{1}>1$, this ratio goes to
infinity as $N\to\infty$ (write e.g. $N^{N}~{}=~{}e^{N\ln N}$) no matter how
large is $C_{2}$.

It is interesting to stress that gravitational interaction behaves at large
distances exactly as $\sim|{x}|^{-3+\varepsilon}$ with $\varepsilon~{}=~{}2$.
Hence we can expect that the matter in the universe does not obey the laws of
thermodynamics and in particular it is not distributed as a homogeneous low-
density gas (and indeed it is really the case!).

#####  2.3.2 Potential with hard core too repuslive at large distances

Consider now a similar case where the pair potential has the same decay as in
(2.11), but now is purely repulsive, i.e. suppose

$V_{3}^{\rm bad}(x)~{}=~{}\begin{cases}+\infty&{\text{i}f}|{x}|\leq a\\\
{1\over|{x}|^{3-\varepsilon}}&{\rm otherwise}\end{cases}$ $None$

Figure 3. The potential $V^{\rm bad}_{3}(|{x}|)$

This is indeed a stable potential (since it is positive!) but not regular. It
will produce with high probability bad configurations in which particles tend
to accumulate at the boundary of $\Lambda$ in a close-packed configuration
hence forming a layer. By an argument identical to the one of case 2,
supposing $\rho\ll\rho_{\rm cp}$, one again shows that “close-packed”
configurations are far more probable than “thermodynamic configurations” (with
particles uniformly distributed in $\Lambda$). Indeed, it is possible to show
that particles interacting via a potential of type (2.14) will tend to leave
the center of the container and accumulate in a layer at the boundary of the
container.

To simplify the calculations let us suppose that the volume $\Lambda$
enclosing the system is a sphere of radius $L$ and let us estimate the
probability of a bad configuration in which the $N$ particles are in
configurations $x_{1},\dots,x_{N}$ which are nearly close-packed (i.e. the
$i^{\rm the}$ particle can move freely inside the sphere $S_{\delta}^{i}$ of
radius $\delta/2$ centered at the vertex $y_{i}$ of the fcc lattice of step
$a+\delta$) occupying a region $\Lambda_{\rm cp}$ which is now a layer stick
at the boundary of the box $\Lambda$ with thickness $\Delta L$. Now, by the
same argument seen above we have that for such configurations (recall that now
the pair potential is everywhere positive) we have

$\displaystyle\sum_{j\in\\{1,2,\dots,N\\}\atop j\neq i}V_{3}^{\rm
bad}(|{x}_{j}-{x}_{i}|)$
$\displaystyle=~{}\sum_{\Delta\subset\Lambda}\sum_{j\in\\{1,2,\dots,N\\}\atop\;j\neq
i,\,{x}_{j}\in\Delta}V_{3}^{\rm bad}(|{x}_{j}-{x}_{i}|)|$
$\displaystyle\leq~{}\sum_{\Delta\subset\Lambda}V_{3}^{\rm bad}(r^{\rm
min}_{\Delta})\sum_{j\in\\{1,2,\dots,n\\}\atop j\neq i,\;{x}_{j}\in\Delta}1$

where this time $r^{\rm min}_{\Delta}$ represent the minimal distance of a
point in the small cube $\Delta$ from $x_{i}$. Now we have that

$\sum_{j\in\\{1,2,\dots,n\\}\atop j\neq i,\;{x}_{j}\in\Delta}1~{}\leq\rho_{\rm
cp}|\Delta|$

Moreover there exists surely an $\eta_{\Delta}$ (depending on $\Delta$) such
that

$\sum_{\Delta\subset\Lambda}V_{3}^{\rm bad}(r^{\rm
max}_{\Delta})|\Delta|\leq(1+\eta_{\Delta})\int_{{x}\in\Lambda_{\rm
cp},~{}|{x}-x_{i}|>a}{1\over|{x}-x_{i}|^{3-\varepsilon}}d{x}$

where $\Lambda_{\rm cp}$ denotes the region (a layer now) inside $\Lambda$
where the $N$ close-packed particles are situated. Now we have that

$\int_{{x}\in\Lambda_{\rm
cp},~{}|{x}-x_{i}|>a}{1\over|{x}-x_{i}|^{3-\varepsilon}}d{x}\leq\int_{{x}\in\Lambda_{\rm
cp}}{1\over|{x}-x_{i}|^{3-\varepsilon}}d{x}\leq\int_{{x}\in\tilde{\Lambda}_{\rm
cp}}{1\over|{x}-x_{i}|^{3-\varepsilon}}d{x}$

where $\tilde{\Lambda}_{\rm cp}$ is a sphere centered at the $x_{i}$ with
volume $|\tilde{\Lambda}_{\rm cp}|=|\Lambda_{\rm cp}|$ (i.e. the sphere
$\tilde{\Lambda}_{\rm cp}$ has the same volume of the layer
$\tilde{\Lambda}_{\rm cp}$ and hence with radius $\tilde{R}=(3|\Lambda_{\rm
cp}|/4\pi))^{1\over 3}$. Therefore

$B_{\varepsilon}(\Lambda_{\rm cp},x_{i})\leq\int_{{x}\in\tilde{\Lambda}_{\rm
cp}}{1\over|{x}-x_{i}|^{3-\varepsilon}}d{x}={4\pi\over\varepsilon}(3/4\pi))^{\varepsilon\over
3}|\Lambda_{\rm cp}|^{\varepsilon\over
3}\leq{4\pi\over\varepsilon}|\Lambda_{\rm cp}|^{\varepsilon\over 3}$

and thus

$\sum_{j\in\\{1,2,\dots,N\\}\atop j\neq i}V_{2}^{\rm
bad}(|{x}_{j}-{x}_{i}|)~{}\leq~{}{4\pi\over\varepsilon}|\Lambda_{\rm
cp}|^{\varepsilon\over 3}\rho_{\rm cp}$

Therefore, recalling that $\rho_{\rm cp}|\Lambda_{\rm cp}|=N$

$U({x}_{1},\dots,{x}_{N})\leq{N\over
2}(1+\eta_{\Delta}){4\pi\over\varepsilon}|\Lambda_{\rm cp}|^{\varepsilon\over
3}\rho_{\rm cp}={N^{1+{\varepsilon\over 3}}\over
2}(1+\eta_{\Delta}){4\pi\over\varepsilon}\rho_{\rm cp}^{2\varepsilon\over 3}$

and hence

$P_{\rm bad}(N)~{}=~{}{1\over
Z_{\Lambda}(\beta,N)}\int_{S^{1}_{\delta}}d{x}_{1}\dots\int_{S^{N}_{\delta}}d{x}_{N}{1\over
N!}e^{-\beta U({x}_{1},\dots,{x}_{N})}\geq$ $\geq{1\over
Z_{\Lambda}(\beta,N)}\left[{4\over 3}\pi\delta^{3}\right]^{N}{1\over
N!}e^{-\beta{N^{1+{\varepsilon\over 3}}\over
2}(1+\eta_{\Delta}){4\pi\over\varepsilon}\rho_{\rm cp}^{2\varepsilon\over 3}}$

On the other hand, for “good” configurations $x_{1},\dots,x_{N}$ in which the
particles are uniformly distributed in $\Lambda$ with density
$\rho=N/|\Lambda|$ (and, say, never lower than $(1-\varepsilon)\rho$ for some
small $\varepsilon$) we have

$\displaystyle\sum_{j\in\\{1,2,\dots,N\\}\atop j\neq i}V_{3}^{\rm
bad}(|{x}_{j}-{x}_{i}|)$
$\displaystyle=~{}\sum_{\Delta\subset\Lambda}\sum_{j\in\\{1,2,\dots,N\\}\atop\;j\neq
i,\,{x}_{j}\in\Delta}V_{3}^{\rm bad}(|{x}_{j}-{x}_{i}|)|$
$\displaystyle\geq~{}\sum_{\Delta\subset\Lambda}V_{3}^{\rm bad}(r^{\rm
max}_{\Delta})\sum_{j\in\\{1,2,\dots,n\\}\atop j\neq i,\;{x}_{j}\in\Delta}1$
$\displaystyle\geq~{}\sum_{\Delta\subset\Lambda}V_{3}^{\rm bad}(r^{\rm
max}_{\Delta})(1-\varepsilon)\rho|\Delta|$

Now, as before

$\sum_{\Delta\subset\Lambda}V_{3}^{\rm bad}(r^{\rm
max}_{\Delta})|\Delta|~{}\geq~{}(1-\eta_{\Delta})\int_{\Lambda:|x-x_{i}|>a}{1\over|x-x_{i}|^{3-\varepsilon}}dx$

But $\int_{\Lambda:|x-x_{i}|>a}{1\over|x-x_{i}|^{3-\varepsilon}}dx$ is minimal
when $x_{i}$ is on the boundary of the sphere $\Lambda$ and we can bound

$\displaystyle\int_{\Lambda:|x-x_{i}|>a}{1\over|x-x_{i}|^{3-\varepsilon}}dx$
$\displaystyle\geq\int_{0}^{2\pi}\int_{0}^{\pi/4}\int_{a}^{L}{r^{2}\over
r^{3-\varepsilon}}drd\theta d\phi$ $\displaystyle={\pi\over
2\varepsilon}\left[L^{\varepsilon}-a^{\varepsilon}\right]$
$\displaystyle\geq~{}{\pi\over 4\varepsilon}L^{\varepsilon}$
$\displaystyle\geq{|\Lambda|^{\varepsilon\over 3}\over 3\varepsilon}$

and thus

$U({x}_{1},\dots,{x}_{N})\geq{N\over
6}{1\over\varepsilon}|\Lambda|^{\varepsilon\over 3}\rho$

and so

$\displaystyle P_{\rm good}(N)$ $\displaystyle={1\over
Z_{\Lambda}(\beta,N)}\int_{\Lambda}d{x}_{1}\dots\int_{\Lambda}d{x}_{N}{1\over
N!}e^{-\beta U({x}_{1},\dots,{x}_{N})}$ $\displaystyle\leq{1\over
Z_{\Lambda}(\beta,N)}|\Lambda|^{N}{1\over N!}e^{-\beta{N\over
6}{1\over\varepsilon}|\Lambda|^{\varepsilon\over 3}\rho}$

Therefore the ratio between the probability of bad and good configurations is
now

$\displaystyle{P_{\rm bad}(N)\over P_{\rm good}(N)}$
$\displaystyle\geq{\left[{4\over 3}\pi\delta^{3}\right]^{N}e^{-\beta{N\over
2}(1+\eta_{\Delta}){4\pi\over\varepsilon}|\Lambda_{\rm cp}|^{\varepsilon\over
3}\rho_{\rm cp}}\over|\Lambda|^{N}e^{-\beta{N\over
6}{1\over\varepsilon}|\Lambda|^{\varepsilon\over 3}\rho}}$
$\displaystyle=~{}\left[{4\pi\delta^{3}\over
3|\Lambda|}\right]^{N}e^{\beta{N^{1+{\varepsilon\over
3}}\over\varepsilon}\left[4\pi(1+\eta_{\Delta})\rho_{\rm cp}^{4\over
3}-{\rho^{4\over 3}\over 6}\right]}$

where recall that $|\Lambda|=\rho N$ and $|\Lambda_{\rm cp}|=\rho_{\rm cp}N$.
Hence we get

$\displaystyle{P_{\rm bad}(N)\over P_{\rm good}(N)}$
$\displaystyle\geq\left[{4\pi\delta^{3}\over
3|\Lambda|}\right]^{N}e^{\beta{N^{1+{\varepsilon\over
3}}\over\varepsilon}\left[4\pi(1+\eta_{\Delta})\rho_{\rm cp}^{4\over
3}-{\rho^{4\over 3}\over 6}\right]}$
$\displaystyle\geq\left[{4\pi\delta^{3}\over
3|\Lambda|}\right]^{N}e^{\beta{N^{1+{\varepsilon\over
3}}\over\varepsilon}\left[\rho_{\rm cp}^{4\over 3}-{\rho^{4\over 3}}\right]}$

which diverges as $N\to\infty$.

Again physics gives us an example of potential such as (2.14). That is, the
purely repulsive Coulomb potential (for which $\varepsilon=2$) between charged
particles with the same charge. Indeed electrons in excess inside a conductor
tend to accumulate at the boundary of the conductor forming layers.

####  2.4 The Ruelle example

In Example 1, $V_{1}^{\rm bad}$ was tempered but not stable. It was a
potential strictly negative at the origin. Therefore a necessary condition for
a pair potential $V(x)$ to be stable is $V(0)\geq 0$. In the two examples of
the previous section we have seen that potentials with a hard-core preventing
that particles accumulate in an arbitrarily small region of $\mathbb{R}^{d}$
but with a two slow decay at large distances (i.e non absolutely integrable
and hence non regular) yield to non thermodynamic behaviors. Failure of
stability however can occur also for regular potential which are strictly
positive in the neighborhood of the origin. We will consider here a very
interesting and surprising example of a potential in $d=3$ dimensions which is
strongly positive near the origin and regular according to Definition 2.2,
that nevertheless is a non stable potential. This subtle example, originally
due to Ruelle, illustrates very well the intuitive fact that the stability
condition is there to avoid the collapse of many particles into a bounded
region of $\mathbb{R}^{d}$ and it also shows the key role played by the
continuum where we have always the possibility to put an arbitrary number of
particles in a small region of $\mathbb{R}^{d}$.

Let $R>0$ and let $\delta>0$ and let

$V^{\rm bad}_{4}({x})~{}=~{}\begin{cases}11&{\rm if}~{}|{x}|<R-\delta\\\
-1&{\rm if}~{}R-\delta\leq|{x}|\leq R+\delta\\\ 0&{\rm otherwise}\end{cases}$
$None$

This is clearly a regular potential (actually it is bounded and finite range:
particles at distances greater than $R+\delta$ do not interact at all).
Neverthless, we will show that this is a non-stable potential by proving that
the grand canonical partition function diverges for such a potential.

Figure 4. A catastrophic potential. The Ruelle potential

Fix an integer $n$ and let $\tilde{x}_{1},\dots,\tilde{x}_{n}$ be sites of a
face-centered cubic lattice in three dimensions with nearest neighborhoods at
distance $R$. We recall that a face-centered cubic lattice is a lattice whose
unit cells are cubes, with lattice points at the center of each face of the
cube, as well as at the vertices.

Let
$B(\tilde{x}_{1},\dots,\tilde{x}_{n})=\\{\\{i,j\\}:|\tilde{x}_{i}-\tilde{x}_{j}|=R\\}$
the set of nearest neighborhood bonds in $\tilde{x}_{1},\dots,\tilde{x}_{n}$
and let $B_{n}~{}=~{}|B(\tilde{x}_{1},\dots,\tilde{x}_{n})|$ the cardinality
of this set. Suppose that $\tilde{x}_{1},\dots,\tilde{x}_{n}$ are arranged in
such way to maximize $B_{n}$, hence in a “close-packed” configuration. We
remind that in the face-centered cubic lattice every site has $12$ nearest
neighborhoods. If we take $\tilde{x}_{1},\dots,\tilde{x}_{n}$ to be close-
packed, then if $n$ is sufficiently large, the number of nearest neighborhoods
bonds are of the order $B_{n}\sim 6n$. In fact each site is the vertex of 12
nearest neighborhoods bonds, and each nearest neighborhood bond is shared
between two sites. If the fixed integer $n$ is chosen sufficiently large then
it is surely possible to find a close-packed configuration
$\tilde{x}_{1},\dots,\tilde{x}_{n}$ in such way that

$B_{n}>{11\over 2}n+\varepsilon$ $None$

for some $\varepsilon>0$. Suppose thus that $n$ is chosen so large such that
the close-packed face centered configuration
$\tilde{x}_{1},\dots,\tilde{x}_{n}$ is such that (2.16) is satisfied. Then,

$\sum_{i=1}^{n}\sum_{j=1}^{n}V_{4}^{\rm
bad}(|\tilde{x}_{i}-\tilde{x}_{j}|)~{}=~{}nV_{4}^{\rm bad}(0)+2B_{n}V_{4}^{\rm
bad}(R)~{}=~{}11n-2B_{n}<-2\varepsilon<0$

Consider now the function

$\Phi:\mathbb{R}^{6n}\to\mathbb{R}:(y_{1},\dots,y_{n},z_{1},\dots,z_{n})\mapsto\sum_{i=1}^{n}\sum_{j=1}^{n}V_{4}^{\rm
bad}(|y_{i}-z_{j}|)$

We have that
$\Phi(\tilde{x}_{1},\dots,\tilde{x}_{n},\tilde{x}_{1},\dots,\tilde{x}_{n})<-2\varepsilon$.
Moreover, if

$|y_{1}-\tilde{x}_{1}|<{\delta\over
2},~{}~{}~{}\dots,~{}~{}~{}|y_{n}-\tilde{x}_{n}|<{\delta\over
2},~{}~{}~{}|z_{1}-\tilde{x}_{1}|<{\delta\over
2},~{}~{}~{}\dots~{}~{}~{}|z_{n}-\tilde{x}_{n}|<{\delta\over 2}$

with $\delta$ being the constant appearing in $\clubsuit$2.10, then also

$\Phi(y_{1},\dots,y_{n},z_{1},\dots,z_{n})=\sum_{i=1}^{n}\sum_{j~{}=~{}1}^{n}V_{4}^{\rm
bad}(|y_{i}-z_{j}|)<-2\varepsilon$ $None$

Let now
$S_{\delta}^{i}~{}=~{}\\{{x}\in\mathbb{R}^{3}:|{x}-\tilde{x}_{i}|<\delta/2\\}$
be the open sphere in $\mathbb{R}^{3}$ with radius $\delta/2$ and center in
$\tilde{x}_{i}$ and define $\Lambda_{\delta}=\cup_{i=1}^{n}S_{\delta}^{i}$
($\Lambda_{\delta}$ is of course a subset of $\mathbb{R}^{3}$).

Let $s$ be a positive integer and define $M_{s}$ as the following subset of
$\mathbb{R}^{3sn}$

$M_{s}~{}=~{}\\{({x}_{1},\dots,{x}_{sn})\in\mathbb{R}^{3sn}:|{x}_{(i-1)s+p}-\tilde{x}_{i}|<\delta/2,~{}~{}p=1,...,s~{}~{}i~{}=1,...,n\\}$

Namely, $({x}_{1},\dots,{x}_{sn})\in M_{s}$ means that the $sn$-uple
$({x}_{1},\dots,{x}_{sn})$ is such that the first $s$ variables
${x}_{1},\dots,{x}_{s}$ of the $sn$-uple are all inside the sphere
$S_{\delta}^{1}$, the variables ${x}_{s+1},\dots,{x}_{2s}$ are all inside the
sphere $S_{\delta}^{2}$, the variables ${x}_{2s+1},\dots,{x}_{3s}$ are all
inside the sphere $S_{\delta}^{3}$, and so on until the last $s$ variables of
the $sn$-uple, which are ${x}_{(n-1)s+1},\dots,{x}_{sn}$, and are all inside
the sphere $S_{\delta}^{n}$.

Now, if $({x}_{1},\dots,{x}_{sn})\in M_{s}$, then,

$\displaystyle U({x}_{1},...,{x}_{sn})$ $\displaystyle=\sum_{1\leq i<j\leq
sn}V_{4}^{\rm bad}(|{x}_{i}-{x}_{j}|)$ $\displaystyle={1\over
2}\left[\sum_{i=1}^{sn}\sum_{j=1}^{sn}V_{4}^{\rm
bad}(|{x}_{i}-{x}_{j}|)-snV_{4}^{\rm bad}(0)\right]$ $\displaystyle={1\over
2}\left[\sum_{p=1}^{s}\sum_{p^{\prime}=1}^{s}\sum_{i=1}^{n}\sum_{j=1}^{n}V_{4}^{\rm
bad}(|{x}_{(i-1)s+p}-{x}_{(j-1)s+p^{\prime}}|)-snV_{4}^{\rm bad}(0)\right]$

Now, for fixed $p$ and $p^{\prime}$ call ${x}_{(i-1)s+p}=y_{i}$ and
${x}_{(j-1)s+p^{\prime}}~{}=~{}z_{j}$. Then, by definition of $M_{s}$ we have
that $|y_{i}-\tilde{x}_{i}|<\delta/2$ and $|z_{j}-\tilde{x}_{j}|<\delta/2$.
Hence by (2.17)

$\displaystyle\sum_{i=1}^{n}\sum_{j~{}=~{}1}^{n}V_{4}^{\rm
bad}(|{x}_{(i-1)s+p}-{x}_{(j-1)s+p^{\prime}}|)$
$\displaystyle=\sum_{i=1}^{n}\sum_{j~{}=~{}1}^{n}V_{4}^{\rm
bad}(|y_{i}-z_{j}|)$ $\displaystyle<-2\varepsilon$

hence we conclude that

$U({x}_{1},\dots,{x}_{sn})<-\left(s^{2}\varepsilon+{11\over
2}sn\right)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm
whenever}~{}~{}({x}_{1},\dots,{x}_{sn})\in M_{s}$

Therefore, if $V_{\delta}$ denote the volume of the sphere of radius
$\delta/2$ in $\mathbb{R}^{3}$ we have

$\Xi_{\Lambda}(\beta,\lambda)~{}=~{}1+\sum_{N=1}^{\infty}{\lambda^{N}\over
N!}\int_{\Lambda}d{x}_{1}\dots\int_{\Lambda}d{x}_{N}~{}e^{-\beta
U({x}_{1},\dots,{x}_{N})}\geq$ $\geq
1+\sum_{s=1}^{\infty}{\lambda^{sn}\over(sn)!}\int_{\Lambda}d{x}_{1}\dots\int_{\Lambda}d{x}_{sn}~{}e^{-\beta
U({x}_{1},\dots,{x}_{sn})}\geq$ $\geq
1+\sum_{s=1}^{\infty}{\lambda^{sn}\over(sn)!}\int_{M_{s}}d{x}_{1}\dots
d{x}_{sn}~{}e^{-\beta U({x}_{1},\dots,{x}_{sn})}~{}\geq$
$\geq~{}\sum_{s=1}^{\infty}{\lambda^{sn}\over(sn)!}V_{\delta}^{sn}~{}e^{{\beta}(s^{2}\varepsilon+{11\over
2}sn)}~{}=~{}\sum_{s=1}^{\infty}{\left[\lambda V_{\delta}e^{{11\beta\over
2}}\right]^{sn}\over(sn)!}\left(e^{\beta\varepsilon}\right)^{s^{2}}~{}=~{}\sum_{s=1}^{\infty}a_{s}$

This last series is a series with positive terms whose term $s^{\rm th}$ term
is given by

$a_{s}~{}=~{}{\left[\lambda V_{\delta}e^{{11\beta\over
2}}\right]^{sn}\over(sn)!}\left(e^{\beta\varepsilon}\right)^{s^{2}}$

In this formula recall that $n$ is a fixed value while $s$ is the variable
integer. It is now easy to show that $\sum_{s=1}^{\infty}a_{s}$ diverges. As a
matter of fact, by the ratio test for positive term series, we have that

$\lim_{s\to\infty}{a_{s+1}\over
a_{s}}~{}=~{}\lim_{s\to\infty}{(sn)!\over[(s+1)n)]!}\left[\lambda
V_{\delta}e^{{11\beta\over
2}}\right]^{n}\left(e^{\beta\varepsilon}\right)^{2s+1}\geq$
$\geq\lim_{s\to\infty}\left[{\lambda V_{\delta}e^{{11\beta\over
2}}\over(s+1)n}\right]^{n}\left(e^{2\beta\varepsilon}\right)^{s}~{}=~{}\left[{\lambda
V_{\delta}e^{{11\beta\over 2}}\over
n}\right]^{n}\lim_{s\to\infty}{\left(e^{2\beta\varepsilon}\right)^{s}\over(s+1)^{n}}~{}=~{}+\infty$

####  2.5 Admissible potentials

Let us start by discussing first three classes of potentials which are both
tempered and stable. 1 - “Repulsive” temperate potentials

$V\geq 0~{}~{}~{}{\rm and}~{}~{}~{}~{}\int_{|x|\geq
r_{0}}V(x)dx<+\infty~{}~{}~{}~{}~{}\mbox{for some $r_{0}\geq 0$}$

2 - Positive type potentials: absolutely integrable potentials which are the
Fourier Transform of a positive function

$V({x})~{}=~{}\int\tilde{V}({\bf k})e^{i{\bf k}\cdot{x}}d^{3}{\bf
k},~{}~{}~{}~{}~{}~{}\tilde{V}({\bf k})\geq 0$

3 - Lennard-Jones Potentials: there exist strictly positive constants $a$,
$C_{1}$, $C_{2}$ and $\varepsilon$ such that

$V({x})\geq{C_{1}\over|{x}|^{3+\varepsilon}}~{}~{}{\rm for}~{}~{}x\leq
a,~{}~{}~{}~{}|V({x})|\leq{C_{2}\over|{x}|^{3+\varepsilon}}~{}~{}{\rm
for}~{}~{}x>a$

The potentials in the class 1, which are automatically stable are called
“repulsive” in a somewhat improper manner: a positive pair potential should be
monotonically decreasing in order to be really purely repulsive. We remark
that tempered potentials with hard core ($V({x})~{}=~{}+\infty$ if $|{x}|\leq
a$) can be included in case 3 or in case 1.

Figure 5. A Lennard-Jones potential

It is immediate to see that a potential of type 1 is admissible. It is
tempered (i.e. it satisfies (2.7)) by definition and it is stable, since from
$V({x})\geq 0$ we get

$U({x}_{1},\dots,{x}_{n})\geq 0~{}~{}~{}~{}~{}~{}\forall
n,{x}_{1},\dots,{x}_{n}$

hence (2.6) is verified. Moreover, since
$\inf_{n,{x}_{1},\dots,{x}_{n}}U({x}_{1},\dots,{x}_{n})=0$, we have that
$B_{V}~{}=~{}0$.

Let us show now that a potential of type 2 is admissible. A positive type
potential $V$ is stable because

$\displaystyle U({x}_{1},\dots,{x}_{n})$ $\displaystyle=\sum_{1\leq i<j\leq
n}V({x}_{i}-{x}_{j})$ $\displaystyle=~{}{1\over 2}\sum_{i\neq
j}V({x}_{i}-{x}_{j})$ $\displaystyle={1\over
2}\sum_{i,j}V({x}_{i}-{x}_{j})-{n\over 2}V({0})$ $\displaystyle={1\over
2}\sum_{i,j}\int e^{{\rm i}{\bf k}\cdot({x}_{i}-{x}_{j})}\tilde{V}({\bf
k})d^{3}{\bf k}-{n\over 2}V(0)$ $\displaystyle=~{}{1\over
2}\int\left[\sum_{i,j}e^{{\rm i}{\bf
k}\cdot({x}_{i}-{x}_{j})}\right]\tilde{V}({\bf k})d^{3}{\bf k}-{n\over 2}V(0)$
$\displaystyle=~{}{1\over 2}\int\left|\sum_{i=1}^{n}e^{{\rm i}{\bf
k}\cdot{x}_{i}}\right|^{2}\tilde{V}({\bf k})d^{3}{\bf k}-{n\over
2}V(0)\geq-{n\over 2}V(0)$

where the last inequality follows from the assumption $\tilde{V}({\bf k})\geq
0$.

Hence the stability condition (2.6) is satisfied by choosing $B~{}=~{}{1\over
2}V(0)$. Potential 2 is tempered since it is supposed to admit Fourier
transform, hence it needs to be absolutely integrable.

The example 3, the Lennard Jones type potential, is of major interest in
applications, since it is the most popular and used by physicists and chemists
to model the interactions between molecules in real gases.

Let us show that Lennard-Jones type potential is admissible. Such a potential
is indeed regular by definition, thus we have just to show that it is also
stable. In a given configuration ${x}_{1},\dots,{x}_{n}$, denote with the
number $r_{\rm min}~{}=~{}\min_{i\neq j}|{x}_{i}-{x}_{j}|$ the minimum
distance between particles.

We distinguish two cases: 1) $r_{\rm min}<{a/2}$; and 2) $r_{\rm
min}\geq{a/2}$. Case 1) $r_{\rm min}<a/2$. Suppose without loss of generality
that $|{x}_{1}-{x}_{2}|=r_{\rm min}$ and all other distances are at distances
greater or equal than $r_{\rm min}$. We have

$\displaystyle U({x}_{1},\dots,{x}_{n})$
$\displaystyle=V({x}_{1}-{x}_{2})+\sum_{j=3}^{n}V({x}_{1}-{x}_{j})+U({x}_{2},\dots,{x}_{n})$
$\displaystyle\geq{C_{1}\over r_{\rm
min}^{3+\varepsilon}}+\sum_{j\in\\{3,\dots,n\\}\atop|{x}_{1}-{x}_{j}|\geq
a}V({x}_{1}-{x}_{j})+U({x}_{2},\dots,{x}_{n})$

Note that the second sum of the last inequality above is over those particles
with index $j\in\\{2,3,\dots,n\\}$ at distance greater or equal to $a$. The
inequality follows from the fact that $V({x}_{1}-{x}_{j})\geq 0$ if
$|{x}_{1}-{x}_{j}|<a$.

We will now get a lower bound for the term
$\sum_{j\in\\{3,\dots,n\\}:~{}|{x}_{1}-{x}_{j}|\geq a}V({x}_{1}-{x}_{j})$.
First note that, since in this sum $|{x}_{1}-{x}_{j}|\geq a$ for all $j$, we
can bound

$\sum_{j\in\\{3,\dots,n\\}:\atop|{x}_{1}-{x}_{j}|\geq
a}V({x}_{1}-{x}_{j})\geq-\sum_{j\in\\{3,\dots,n\\}:\atop|{x}_{1}-{x}_{j}|\geq
a}{C_{2}\over|{x}_{1}-{x}_{j}|^{3+\varepsilon}}$

We then proceed as follows. Draw around each ${x}_{j}$ a cube $Q_{j}$ with
side $r_{\rm min}/{\sqrt{1}2}$ (in such way that its maximal diagonal is
$r_{\rm min}/2$) with ${x}_{j}$ being the vertex farthest away from ${x}_{1}$.
Since any two points among ${x}_{3},\dots,{x}_{n}$ are at mutual distances
$\geq r_{\rm min}$ the cubes so constructed do not overlap. Moreover, if we
consider the open sphere
$S_{1}~{}=~{}\\{{x}\in\mathbb{R}^{3}:|{x}-{x}_{1}|<{a\over 2}\\}$ and denote
by $S_{1}^{c}$ its complementar in $\mathbb{R}^{3}$, then by construction
(recall that $r_{\rm min}<a/2$ and $|x_{1}-x_{j}|\geq a$) all cubes
$Q_{j}\subset S_{1}^{c}$, i.e., they lay outside the open sphere with center
${x}_{i}$ and radius $a/2$. Furthermore, we have

${1\over|{x}_{1}-{x}_{j}|^{3+\varepsilon}}\leq{(\sqrt{12})^{3}\over{r_{\rm
min}^{3}}}\int_{Q_{j}}{1\over|{x}_{1}-{x}|^{3+\varepsilon}}d{x}$

recall in fact that the cube $Q_{j}$ is chosen in such way that
$|{x}-{x}_{1}|\leq|{x}_{1}-{x}_{j}|$ for all ${x}\in Q_{j}$. Therefore

$\displaystyle\sum_{j\in\\{3,\dots,n\\}:\atop|{x}_{1}-{x}_{j}|\geq
a}^{n}V({x}_{1}-{x}_{j})$ $\displaystyle\geq-C_{2}{(\sqrt{12})^{3}\over{r_{\rm
min}^{3}}}\sum_{j\in\\{3,\dots,n\\}:\atop|{x}_{1}-{x}_{j}|\geq
a}^{n}\int_{Q_{j}}{1\over|{x}_{1}-{x}|^{3+\varepsilon}}d^{3}{x}$
$\displaystyle=-C_{2}{(\sqrt{12})^{3}\over{r_{\rm
min}^{3}}}\int_{\cup_{j}Q_{j}}{1\over|{x}_{1}-{x}|^{3+\varepsilon}}d^{3}{x}$
$\displaystyle\geq-C_{2}{(\sqrt{12})^{3}\over{r_{\rm
min}^{3}}}\int_{|{x}-{x}_{1}|\geq
a/2}{1\over|{x}-{x}_{1}|^{3+\varepsilon}}d^{3}{x}$
$\displaystyle=-C_{2}{(\sqrt{12})^{3}\over{r_{\rm min}^{3}}}\int_{|y|\geq
a/2}{1\over|y|^{3+\varepsilon}}d^{3}y$
$\displaystyle={C_{2}K_{a}(\varepsilon)\over r_{\rm min}^{3}}$

where

$K_{a}(\varepsilon)~{}=~{}{(\sqrt{12})^{3}}{4\pi
2^{\varepsilon}\over\varepsilon a^{\varepsilon}}$

So

$U({x}_{1},\dots,{x}_{n})~{}\geq~{}{C_{1}\over r_{\rm
min}^{3+\varepsilon}}-{C_{2}K_{a}(\varepsilon)\over r_{\rm
min}^{3}}+U({x}_{2},\dots,{x}_{n})$

I.e.

$U({x}_{1},\dots,{x}_{n})\geq-C_{\varepsilon}+U({x}_{2},\dots,{x}_{n})$

where $-C_{\varepsilon}$ is the minimum of the function $F(x)={C_{1}\over
x^{3+\varepsilon}}-{C_{2}K_{a}(\varepsilon)\over x^{3}}$ for $x>0$. Therefore,
iterating this formula we get

$U({x}_{1},\dots,{x}_{n})\geq-C_{\varepsilon}n$

Case 2) $r_{\rm min}\geq a/2$. In this case we write

$U({x}_{1},\dots,{x}_{n})~{}=~{}{1\over
2}\sum_{i=1}^{n}\sum_{j\in\\{1,2,\dots,n\\}:\,\,j\neq i}V({x}_{i}-{x}_{j})$

and we will get an estimate $\sum_{j\in\\{1,2,\dots,n\\},\,j\neq
i}V({x}_{i}-{x}_{j})$. First we bound

$\sum_{j\in\\{1,2,\dots,n\\}\atop j\neq
i}V({x}_{i}-{x}_{j})\geq-\sum_{j\in\\{1,2,\dots,n\\}\atop|x_{i}-x_{j}|\geq
a}{C_{2}\over|{x}_{i}-{x}_{j}|^{3+\varepsilon}}$

We then proceed analogously as before. This time, for fixed $i$, we can draw
around each ${x}_{j}$ a cube $Q_{j}$ with side $a/{\sqrt{4}8}$ (such that the
maximal diagonal of $Q_{j}$ is $a/4$) with ${x}_{j}$ being the vertex farthest
away from ${x}_{i}$. Since any two points among ${x}_{1},\dots,{x}_{n}$ are at
mutual distances $\geq a/2$ the cubes so constructed do not overlap. Moreover
if we consider the open sphere
$S_{i}~{}=~{}\\{{x}\in\mathbb{R}^{3}:|{x}-{x}_{i}|<{a\over 2}\\}$ and denote
by $S_{i}^{c}$ its complementar in $\mathbb{R}^{3}$, then by construction
$\cup_{j\neq i}Q_{j}\subset S_{i}^{c}$. In other words all cubes $Q_{j}$ lay
outside the open sphere with center ${x}_{i}$ and radius $a/2$. Furthermore we
have

${1\over|{x}_{i}-{x}_{j}|^{3+\varepsilon}}\leq{(\sqrt{48})^{3}\over
a^{3}}\int_{Q_{j}}{1\over|{x}_{i}-{x}|^{3+\varepsilon}}d^{3}{x}$

recall in fact that the cube $Q_{j}$ is chosen in such way that
$|{x}-{x}_{i}|\leq|{x}_{i}-{x}_{j}|$ for all ${x}\in Q_{j}$.

Therefore

$\displaystyle\sum_{j\in\\{1,2,\dots,n\\}\atop|x_{1}-x_{j}|\geq
a}V({x}_{i}-{x}_{j})$ $\displaystyle\geq~{}-C_{2}{(\sqrt{48})^{3}\over
a^{3}}\sum_{j\in\\{1,2,\dots,n\\}\atop x_{1}-x_{j}|\geq
a}\int_{Q_{j}}{1\over|{x}_{i}-{x}|^{3+\varepsilon}}d^{3}{x}$
$\displaystyle=-C_{2}{(\sqrt{48})^{3}\over a^{3}}\int_{\cup_{j\neq
i}Q_{j}}{1\over|{x}_{i}-{x}|^{3+\varepsilon}}d^{3}{x}$ $\displaystyle\geq-
C_{2}{(\sqrt{48})^{3}\over a^{3}}\int_{|{x}-x_{i}|\geq{a\over
2}}{1\over|{x}_{i}-{x}|^{3+\varepsilon}}d^{3}{x}$
$\displaystyle\geq~{}-{\bar{K}_{a}}(\varepsilon)$

where

$\bar{K}_{a}(\varepsilon)~{}=~{}C_{2}{(\sqrt{48})^{3}\over
a^{3}}\int_{|{x}-x_{i}|\geq a/2}{1\over|{x}_{i}-{x}|^{3+\varepsilon}}d^{3}{x}$

Hence

$\sum_{j\in\\{1,2,\dots,n\\}\atop j\neq
i}V({x}_{i}-{x}_{j})~{}\geq~{}-{\bar{K}_{a}}(\varepsilon)$ $None$

and

$U({x}_{1},\dots,{x}_{n})~{}\geq~{}-{\bar{K}_{a}(\varepsilon)\over 2}n$

#####  2.5.1 Basuev Criteria

We now present two very efficient criteria for stability of tempered
potentials first proposed by Basuev [2]. To do this we need to itroduce the
following notation. Given a potential $V(x)$ let

$V(a)=\inf_{|x|=a}V(x)$ $None$

Given $n\in\mathbb{N}$, let us consider the set of configurations defined by

$\mathcal{S}_{n}(a)=\\{(x_{1},\dots,x_{n})\in\mathbb{R}^{dn}:~{}|x_{i}-x_{j}|>a,~{}~{}~{}{\rm
for}~{}1\leq i<j\leq n\\}$

I.e. $\SS_{n}(a)$ is formed by all configurations of $n$ particles in
$\mathbb{R}^{dn}$ such that any two particles are at mutual distance greater
than $a$.

###### Definition 2.4

A potential $V(x)$ is called Basuev if there exists $a>0$ such that $V(a)>0$,
$V(x)\geq V(a)$ for all $|x|\leq a$ and

$V(a)\geq\mu(a)$ $None$

where

$\mu(a)=\sup_{n\in\mathbb{N},\atop{(x_{1},\dots,x_{n})\in\mathcal{S}_{n}(a)}}\sum_{i=1}^{n}V^{-}(x_{i})$
$None$

with

$V^{-}(x)=\max\\{-V(x),0\\}.$

V(x) is called strongly Basuev if

$V(a)\geq 2\mu(a)$ $None$

We now prove the following theorem.

###### Theorem 2.1

Let $V(x)$ be a Basuev pair potential with Basuev constants $a$, $V(a)$ and
$\mu(a)$ as in definition 2.4. Then $V(x)$ is stable with stability constant
$B_{V}$ such that

$B_{V}\leq\mu(a)$

Remark. Since $V(|x|)$ is translational invariant, observe that for any
$x_{0}\in\mathbb{R}^{d}$ it holds that

$\sup_{n\in\mathbb{N},\atop{(x_{1},\dots,x_{n})\in\mathbb{R}^{dn}\atop|x_{i}-x_{j}|>a}}\sum_{i=1}^{n}V^{-}(x_{0}-x_{i})=\mu(a)$
$None$

Proof.  Let $n\in\mathbb{N}$ be fixed and let us first consider the case in
which we have a configuration of $n$ particles in positions
$x_{1},\dots,x_{n}$ such that all $n$ particles are mutually at distance
greater than $a$, i.e let us consider first the set of configurations in
$\SS_{n}(a)$. We have that, for any $n\in\mathbb{N}$

$\displaystyle\inf_{(x_{1},\dots,x_{n})\in\SS_{n}(a)}{1\over
n}U(x_{1},\dots,x_{n})$
$\displaystyle=\inf_{(x_{1},\dots,x_{n})\in\SS_{n}(a)}{1\over n}\sum_{1\leq
k<s\leq n}V(x_{k}-x_{s})$
$\displaystyle=\inf_{(x_{1},\dots,x_{n})\in\SS_{n}(a)}{1\over
2n}\sum_{k=1}^{n}\sum_{s\neq k}V(x_{k}-x_{s})$ $\displaystyle\geq-{1\over
2n}\sum_{k=1}^{n}\sup_{(x_{1},\dots,x_{n})\in\SS_{n}(a)}\sum_{s\neq
k}V^{-}(x_{k}-x_{s})$ $\displaystyle\geq-{1\over
2n}\sum_{k=1}^{n}\sup_{s\in\mathbb{N}\atop{(x_{1},\dots,x_{s})\in\SS_{n}(a)}}\sum_{j=1}^{s}V^{-}(x_{j})=-{\mu(a)\over
2}$

where in the last inequality we have used (2.23) and (2.21). We can therefore
limit ourselves to configurations in which there are particles at distance $a$
or smaller than $a$. Consider thus a configuration $(x_{1},\dots,x_{n})$ such
that there exists $\\{i,j\\}\subset[n]$ such that $|x_{i}-x_{j}|\leq a$. Thus
there is a particle, which, without loss of generality, we can assume to be
the particle indexed by 1 at position $x_{1}=0$ (i.e. at the origin), which
has the maximum number of particles among $x_{2},\dots,x_{n}$ at distance less
than or equal $a$. We have to estimate, for any $n\in\mathbb{N}$

$U(x_{1},\dots,x_{n})=E_{1}+U(x_{2},\dots,x_{n})$

where

$E_{1}=\sum_{j=2}^{n}V(x_{j})$

Say that the number of these particles close to $x_{1}$ less or equal to $a$
is $l$ (clearly $l\geq 1$ by assumption). The energy $E_{1}$ of the particle
at position $x_{1}$ is thus

$E_{1}\geq lV(a)-\sum_{k\in[n]\atop|x_{k}|>a}V^{-}(x_{k})$ $None$

To bound the sum

$\sum_{k\in[n]\atop|x_{k}-x_{1}|>a}V^{-}(x_{k}))$

observe that we are supposing that each particle has at most $l$ other
particles at distance less or equal than $a$. Thus take the $k\in[n]$ such
that $V^{-}(x_{k})$ is maximum. Again, without loss of generality we can
suppose $k=2$. In the sphere with center $x_{2}$ and radius $a$ there are at
most $l+1$ particles (the particle at position $x_{2}$, for which the value
$V^{-}(|x_{2}|)$ is maximum plus at most $l$ other particles, say
$x_{i_{1}},\dots,x_{i_{l}}$, for which $V^{-}(|x_{i_{s}}|)\leq V^{-}(x_{2})$
with $s\in[l]$). Hence

$\sum_{k\in[n]\atop|x_{k}-x_{1}|>a}V^{-}(|x_{k}|)\leq(l+1)V^{-}(x_{2})+\sum_{k\in[n]\atop|x_{k}-x_{1}|>a,|x_{k}-x_{2}|>a}V^{-}(x_{k})$

Now we have to control the sum

$\sum_{k\in[n]\atop|x_{k}-x_{1}|>a,|x_{k}-x_{2}|>a}V^{-}(|x_{k}|)$

Note that in this sum all particles are at distance greater than $a$ for the
particle in $x_{1}$ and also from particle at position $x_{2}$ moreover each
of the particle in the sum has at most $l$ particles at distance less or equal
than $a$. Suppose without loss of generality that the particle at position
$x_{3}$ is such that

$V^{-}(x_{3})=\max_{k\in[n]\atop|x_{k}-x_{1}|>a,|x_{k}-x_{2}|>a}V^{-}(x_{k})$

and this particle at $x_{3}$ has at most $l$ particles at distance less or
equal than $a$. Therefore

$\sum_{k\in[n]\atop|x_{k}-x_{1}|>a,|x_{k}-x_{2}|>a}V^{-}(x_{k})\leq(l+1)V^{-}(x_{3})+\sum_{k\in[n]\atop|x_{k}-x_{1}|>a,|x_{k}-x_{2}|>a,\,|x_{k}-x_{3}|>a}V^{-}(x_{k})$

Iterating we get

$\sum_{k\in[n]\atop|x_{k}-x_{1}|>a}V^{-}(x_{k})\leq(l+1)\sum_{k\in[n]\atop|x_{i}-x_{j}|>a}V^{-}(x_{k})$
$None$

where now in the sum in the r.h.s of $\clubsuit$lpu all pairs of particles are
at distance greater than $a$ to each other. Therefore, recalling definition
(2.21) we have that

$\sum_{k\in[n]\atop|x_{i}-x_{j}|>a}V^{-}(x_{k})\leq\mu(a)$

so that

$\sum_{k\in[n]\atop|x_{k}-x_{1}|>a}V^{-}(x_{k})\leq(l+1)\mu(a)$

Therefore, recalling (2.24), we have that

$E_{1}\geq lV(a)-(l+1)\mu(a)=l(V(a)-\mu(a))-\mu(a)\geq-\mu(a)$

So we have obtained

$U(x_{1},\dots,x_{n})\geq-\mu(a)+U(x_{2},\dots,x_{n})$

Iterating we get

$U(x_{1},\dots,x_{n})\geq-n\mu(a)$

$\Box$ Example. Let us consider the potential

$V({x})~{}=~{}\begin{cases}A&{\rm if}~{}|{x}|\leq R\\\ -1&{\rm
if}~{}R<|{x}|\leq R+\delta\\\ 0&{\rm otherwise}\end{cases}$ $None$

Let us prove that $V(x)$ is stable if $A\geq 12$ and $\delta$ sufficiently
small.

It is known that the maximum number of points that can be put on the surface
of the sphere $S(R)$ of radius $R$ in such way that the distance between any
pair of such points is greater or equal $R$ is 12. Let us now show that if
$\delta$ is sufficiently small 12 is also the maximum number of points that
can be fitted in the layer $S_{\delta}(R)=\\{x\in\mathbb{R}^{3}:R\leq|x|\leq
R+\delta\\}$. Indeed, suppose by absurd that for any $\delta>0$ one can find
13 points on $S_{\delta}$ such that the distance between any pair of these
points is greater or equal than $R$. Then by choosing $\delta={1\over n}$, for
$n\in\mathbb{N}$ one can construct 13 sequences $x_{n}^{1},\dots,x_{n}^{13}$
all contained in $S_{\delta}$ such that, for any $n$ and for any
$i,j\in\\{1,2,\dots,13\\}$ it holds that $|x_{n}^{i}-x_{n}^{j}|\geq R$ since
$S_{\delta}$ is compact and $\lim_{n\to\infty}S_{\delta}(R)=S(R)$, for each of
the sequences $x^{i}_{n}$ there exists a subsequence convergent to some point
$y^{i}\in S(R)$ such that for any pair $|y^{i}-y^{j}|\geq R$. So we have found
13 points of the sphere $S(R)$ of radius $R$ all at distance from one to
another greater that $R$, contradicting the statement that the maximal number
of such points is 12. This means that the maximum number of points we can put
on the layer $S_{\delta}$ in such way that the distance between any pair of
such points is greater or equal $R$ is 12 and therefore the potential of
Figure 4 with 11 is replaced by any number $A\geq 12$ and $\delta$ is
sufficiently small satisfies the hypothesis of Theorem 2.1 and hence is
stable.

###### Theorem 2.2

Let $V(x)$ be a strongly Basuev and tempered pair potential, with costants
$a$, $V(a)$ and $\mu(a)$ as in definition 2.4. Then $V(x)$ is stable with
stability constant $B_{V}\leq{1\over 2}\mu(a)$. Moreover the representation

$V(x)=V_{a}(x)+K_{a}(x)$

with

$V_{a}(x)=\begin{cases}V(a)&{\rm if}~{}|x|\leq a\\\ V(x)&{\rm
if}~{}|x|>a\end{cases}$ $None$

and

$K_{a}(x)=\begin{cases}V(x)-V(a)&{\text{i}f}~{}|x|\leq a\\\
0&{\text{i}f}~{}|x|>a\end{cases}$ $None$

is such that the potential $V_{a}(x)$ defined in (2.27) is also stable and it
has the same stability constant $B_{V}$ of the full potential $V(x)$ and the
potential $K_{a}(x)$ is positive and supported in $[0,a]$.

Proof.  The thesis is trivial if $V^{-}(x)=0$ (i.e. if $V(x)$ is purely
repulsive). So we may assume that $V^{-}(x)\neq 0$. For any
$(x_{1},\dots,x_{n})\in\mathbb{R}^{nd}$ and any $i\in[n]$, let

$E_{i}(x_{1},\dots,x_{n})=\sum_{j\in[n]:\,j\neq i}V(x_{i}-x_{j})$

so that $U(x_{1},\dots,x_{n})={1\over
2}\sum_{i\in[n]}E_{i}(x_{1},\dots,x_{n})$. Note that when $V^{-}\neq 0$ then
surely $B>0$ and thus configurations $x_{1},\dots,x_{n}$ which minimize the
energy must be searched between those such that $U(x_{1},\dots,x_{n})<0$. Let
now $(x_{1},\dots,x_{n})\in\mathbb{R}^{nd}$ be a configuration such that
$U(x_{1},\dots,x_{n})<0$ but there is a particle in position say $x_{1}$
(without loss of generality) such that $E_{1}(x_{1},\dots,x_{n})\geq 0$. Then

$U(x_{1},\dots,x_{n})=E_{1}(x_{1},\dots,x_{n})+U(x_{2},\dots,x_{n})$

and, since we are assuming $E_{1}(x_{1},\dots,x_{n})\geq 0$ and
$U(x_{1},\dots,x_{n})<0$, we have that $U(x_{2},\dots,x_{n})<0$ and

$U(x_{1},\dots,x_{n})\geq U(x_{2},\dots,x_{n})$

i.e.

$-U(x_{1},\dots,x_{n})\leq-U(x_{2},\dots,x_{n})$

with both sides of the last inequality positive. Therefore, since ${1\over
n}<{1\over n-1}$,

$-{1\over n}U(x_{1},\dots,x_{n})<-{1\over n-1}U(x_{2},\dots,x_{n})$

Thus the configuration $(x_{2},\dots,x_{n})$ produces a value $-{1\over
n-1}U(x_{2},\dots,x_{n})$ which is nearer to $B_{V}$ than
$-U(x_{1},\dots,x_{n})/n$. Whence we can look for minimal energy
configurations $(x_{1},\dots,x_{n})$ limiting ourselves to those
configurations in which the energy per particle $E_{i}(x_{1},\dots,x_{n})$ is
negative for all $i\in[n]$.

Now let us consider the system of particles interacting via the pair potential
$V_{a}(x)$ defined in (2.27) and let us assume that conditions (2.20) and
(2.22) hold. Note first that, recalling definition (2.27),
$V_{a}^{-}(|x|)=\max\\{0,-V_{a}(x)\\}=V^{-}(x)$. Consider then a configuration
$(x_{1},\dots,x_{n})$ such that there exists $\\{i,j\\}\subset[n]$ such that
$|x_{i}-x_{j}|\leq a$. Thus there is at least a particle, (which, without loss
of generality, we can assume to be the particle indexed by 1 at position
$x_{1}$), which has the maximum number of particles among $x_{2},\dots,x_{n}$
at distance less than or equal than $a$. Say that the number of these
particles close to $x_{1}$ less or equal to $a$ is $l$ (clearly $l\geq 1$ by
assumption). The energy $E_{1}$ of the particle at position $x_{1}$ can thus
be estimated as follows.

$E_{1}(x_{1},\dots,x_{n})\geq
lV(a)-\sum_{k\in[n]\atop|x_{k}-x_{1}|>a}V^{-}(x_{k})$

To control the sum

$\sum_{k\in[n]\atop|x_{k}-x_{1}|>a}V^{-}(x_{k}))$

observe that we are supposing that each particle has at most $l$ other
particles at distance less or equal than $a$. Thus take the $k\in[n]$ such
that $V^{-}(x_{k})$ is maximum. Again, without loss of generality we can
suppose $k=2$. In the sphere with center $x_{2}$ and radius $a$ there are at
most $l+1$ particles (the particle at position $x_{2}$, for which the value
$V^{-}(|x_{2}|)$ is maximum plus at most $l$ other particles). Hence

$\sum_{k\in[n]\atop|x_{k}-x_{1}|>a}V^{-}(x_{k}|)\leq(l+1)V^{-}(x_{2})+\sum_{k\in[n]\atop|x_{k}-x_{1}|>a,|x_{k}-x_{2}|>a}V^{-}(x_{k})$

Now we have to control the sum

$\sum_{k\in[n]\atop|x_{k}-x_{1}|>a,|x_{k}-x_{2}|>a}V^{-}(x_{k})$

Note that in this sum all particles are at distance greater than $a$ for the
particle in $x_{1}$ and also from the particle at position $x_{2}$. Moreover,s
each of the particle in the sum has at most $l$ particles at distance less or
equal than $a$. Suppose without loss of generality that the particle at
position $x_{3}$ is such that

$V^{-}(x_{3})=\max_{k\in[n]\atop|x_{k}-x_{1}|>a,|x_{k}-x_{2}|>a}V^{-}(x_{k})$

and this particle at $x_{3}$ has at most $l$ particles at distance less or
equal than $a$. Therefore

$\sum_{k\in[n]\atop|x_{k}-x_{1}|>a,|x_{k}-x_{2}|>a}V^{-}(x_{k})\leq(l+1)V^{-}(x_{3})+\sum_{k\in[n]\atop|x_{k}-x_{1}|>a,|x_{k}-x_{2}|>a,\,|x_{k}-x_{3}|>a}V^{-}(x_{k})$

Iterating we get

$\sum_{k\in[n]\atop|x_{k}-x_{1}|>a}V^{-}(x_{k})\leq(l+1)\sum_{k\in[n]\atop|x_{i}-x_{j}|>a}V^{-}(x_{k})$
$None$

where now in the sum in the r.h.s of (2.29) all pairs of particles are at
distance greater than $a$ to each other. Therefore, recalling definition
(2.21) we have that

$\sum_{k\in[n]\atop|x_{i}-x_{j}|>a}V^{-}(x_{k})\leq\mu(a)$

and hence

$E_{1}(x_{1},\dots,x_{n})\geq lV(a)-(l+1)\mu(a)$

I.e. we have that $E_{1}>0$ whenever

$V(a)>{l+1\over l}\mu(a)$

Using now assumption (2.22) and since ${l+1\over l}<2$ we get

$E_{1}(x_{1},\dots,x_{n})>0$

The conclusion is that if a configuration $(x_{1},\dots,x_{n})$ is such that
some particles are at distance less or equal than $a$ then there is at least a
particle whose energy is positive. Hence the minimal energy configurations for
$V_{a}(x)$ must be searched among those configurations in which all particles
are at distance greater than $a$ from each other. But for these configurations
$V_{a}(|x|)=V(|x|)$ which implies that $V_{a}$ and $V$, if stable, have the
same stability constant $B$ (and hence also the same $\bar{B}$). It is now
easy to see that $V_{a}(|x|)$ is stable. Indeed observe that, for any
configuration $(x_{1},\dots,x_{n})$ for which particles are at distance
greater than $a$ from each other we have

$U_{a}(x_{1},\dots,x_{n})={1\over 2}\sum_{i=1}^{n}\sum_{j\in[n]\atop j\neq
i}V(x_{i}-x_{j})\geq-{1\over 2}n\mu(a)$

which implies that $V_{a}(x)$ is stable with stability constant
$B\leq{\mu(a)\over 2}$. $\Box$ We have shown above that a Lennard-Jones type
potential $V(|x|)$ is stable. Here below we prove that a Lennard-Jones type
potential is also Basuev.

###### Theorem 2.3

Let $V(x)$ be a pair potential on $\mathbb{R}^{d}$ such that there exist
constants $w,r_{1},r_{2}>0$, with $r_{1}\leq r_{2}$, and non-negative
monotonic decreasing functions $\xi(|x|)$, $\eta(|x|)$ such that

$V(x)\begin{cases}\geq\xi(|x|)&{\rm if}~{}|x|\leq r_{1}\\\ \geq-w&{\rm
if}~{}r_{1}<|x|<r_{2}\\\ \geq-\eta(|x|)&{\rm if}~{}|x|\geq r_{2}\end{cases}$
$None$

with

$\lim_{x\to 0}\xi(x)x^{d}=+\infty$ $None$

and

$\int_{|x|\geq r_{2}}\eta(|x|)dx<+\infty$ $None$

Then $V(x)$ is strongly Basuev (i.e. it satisfies Theorem 2.2).

Proof.  By Theorem 2.2 we just need to show that there exists $a$ such that
(2.20) and (2.22) are satisfied. Fix $a\in(0,r_{1})$, let
$\bar{w}=\max\\{w,\eta(r_{2})\\}$

$\bar{\eta}(|x|)=\begin{cases}\eta(|x|)&{\text{i}f}~{}|x|>r_{2}\\\
\bar{w}&{\text{i}f}~{}|x|\leq r_{2}\end{cases}$

Then, by construction $\bar{\eta}(|x|)$ is monotonic decreasing and such that
$\int_{\mathbb{R}^{d}}\bar{\eta}(|x|)\leq\infty$. Moreover by conditions
(2.30) we have that

$V^{-}(x)\leq\bar{\eta}(|x|)$

Hence, recalling (2.21) and considering also that, since we took
$a\in(0,r_{1})$, by hypothesis $V^{-}(|x|)=0$ for all $|x|\leq a$, we have

$\mu(a)\leq\sup_{n\in\mathbb{N},~{}(x_{1},\dots,x_{n})\in\mathbb{R}^{dn}\atop|x_{i}-x_{j}|>a,~{}|x_{i}|>a}\sum_{i=1}^{n}\bar{\eta}(|x_{i}|)$

To estimate from above $\sum_{i=1}^{n}\bar{\eta}(|x_{i}|)$, having in mind
that all particles are at mutual distances greater than $a$ and are at
distance greater than $a$ from the origin, we proceed as follows. We draw for
each $x_{j}$ a hypercube $Q_{j}$ with side $a/2\sqrt{d}$ (such that the
maximal diagonal of $Q_{j}$ is $a/2$) in such a way that $x_{j}$ is a vertex
of the cube $Q_{j}$ and at the same time is the point $x\in Q_{j}$ which is
the farthest away from the origin $0$. Since any two points among
$x_{1},\dots,x_{n}$ are at mutual distances $\geq a$ the cubes so constructed
do not overlap. Furthermore, using the fact that $\bar{\eta}(|x|)$ is
monotonic decreasing, we have

$\bar{\eta}(|x_{j}|)\leq{(4d)^{d\over 2}\over
a^{d}}\int_{Q_{j}}\bar{\eta}(|x|)dx$

recall in fact that the cube $Q_{j}$ is chosen in such way that
$|x|\leq|x_{j}|$ for all $x\in Q_{j}$. Therefore

$\displaystyle\sum_{i=1}^{n}\bar{\eta}(|x_{i}|)$
$\displaystyle\leq{(4d)^{d\over 2}\over
a^{d}}\sum_{i=1}^{n}\int_{Q_{i}}\bar{\eta}(|x|)dx$
$\displaystyle={(4d)^{d\over 2}\over
a^{d}}\int_{\cup_{i}Q_{i}}\bar{\eta}(|x|)dx$ $\displaystyle\leq{{(4d)^{d\over
2}\over a^{d}}}\int_{\mathbb{R}^{d}}\bar{\eta}(|x|)dx$
$\displaystyle={C_{d}\over a^{d}}$

where $C_{d}=~{}{(4d)^{d\over 2}}\int_{\mathbb{R}^{d}}\bar{\eta}(|x|)dx$.
Hence we get

$\mu(a)\leq{C_{d}\over a^{d}}$

Now, in view of condition (2.31), we can always choose $a$ such that
$\xi(a)a^{d}>2C_{d}$. Thus we get

$\xi(a)a^{d}>2C_{d}~{}~{}~{}\Longrightarrow~{}~{}\xi(a)>2{C_{d}\over
a^{d}}~{}~{}~{}\Longrightarrow~{}~{}~{}V(a)>2{C_{d}\over
a^{d}}~{}~{}~{}\Longrightarrow~{}~{}~{}V(a)>2\mu(a)$

$\Box$ Exercise: prove that a Lennard-Jones pair potential satisfies Theorem
2.3 and hence Theorem 2.2 (thus a Lennard-Jones potential can be written as
the sum of a positive potential plus an absolutely integrable and stable
potential with stability constant $B_{V}$ equal to the one of the full
potential).

####  2.6 The infinite volume limit

We will now start to consider the problem of the existence of the
thermodynamic limit for the pressure of a system of particles in the grand
canonical ensemble interacting via a pair potential stable and tempered. The
mathematical problem is to show the existence of the infinite volume pressure
defined as

$\beta
p(\beta,\lambda)~{}=~{}\lim_{\Lambda\to\infty}{1\over|\Lambda|}\ln\Xi_{\Lambda}(\beta,\lambda)$
$None$

where we recall that $|\Lambda|$ denotes the volume of $\Lambda$.

First of all we need to give a mathematical meaning to the notation
$\Lambda\to\infty$ in (2.33). We know that $\Lambda$ is a finite region of
$\mathbb{R}^{3}$ and it can tend to infinity (namely its volume tends to
infinity) in various ways. For instance, $\Lambda$ could be a cylinder of
fixed base $A$ and height $h$ and we could let $h\to\infty$. I.e. like a cigar
with increasing length. It is obvious that such a system (particularly if $A$
is very small) is not expected to have a thermodynamic behavior even if
$h\to\infty$.

Thus $\Lambda\to\infty$ cannot simply be $|\Lambda|\to\infty$, since we want
to exclude cases like “the cigar”. We need thus that $\Lambda\to\infty$
roughly in such way that $\Lambda$ is big in every direction, e.g. a sphere of
increasing radius, a cube of increasing size etc. We will review the following
definitions

###### Definition 2.5

$\Lambda$ is said to go to infinity in the sense of Van Hove if the following
occurs.

Cover $\Lambda$ with small cubes of size $a$ and let $N_{+}(\Lambda,a)$ the
number of cubes with non void intersection with $\Lambda$ and
$N_{-}(\Lambda,a)$ the number of cubes strictly included in $\Lambda$. Then
$\Lambda\to\infty$ in the sense of Van Hove if

$N_{-}(\Lambda,a)\to\infty,~{}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}{N_{-}(\Lambda,a)\over N_{+}(\Lambda,a)}\to
1~{}~{}~{}~{}~{}~{}~{}~{}\forall a$

See figure 7.

As an example, it is easy to show that if $\Lambda\to\infty$ as in the case of
the “cigar” seen before, then $\Lambda$ is not tending to infinity in the
sense of Van Hove. Consider thus, for sake of simplicity in the plane, a
rectangle $R$ with sizes $l$ fixed and $t$ variable and going to infinity. Let
consider squares of size $a$ to cover the rectangle. Then

$N_{-}(R,a)~{}=~{}{l\over a}{t\over
a},~{}~{}~{}~{}~{}~{}~{}~{}N_{+}(R,a)~{}=~{}{l\over a}{t\over a}+2{t\over
a}+2{l\over a}$

thus

$\lim_{t\to\infty}{N_{-}(R,a)\over N_{+}(R,a)}~{}=~{}{l\over l+2a}\neq 1$

On the other hand, let us check that a rectangle of sizes $(\sqrt{t},t)$ and
tends to infinity, as $t\to\infty$, in the sense of Van Hove. As a matter of
fact

$N_{-}(R,a)~{}=~{}{\sqrt{t}\over a}{t\over
a},~{}~{}~{}~{}~{}~{}~{}~{}N_{+}(R,a)~{}=~{}{\sqrt{t}\over a}{t\over
a}+2{t\over a}+2{\sqrt{t}\over a}$

thus

$\lim_{t\to\infty}{N_{-}(R,a)\over
N_{+}(R,a)}~{}=~{}\lim_{t\to\infty}{t\sqrt{t}\over
t\sqrt{t}+2at+2a\sqrt{t}}~{}=~{}1$

We now give a second and more stringent definition of $\Lambda\to\infty$. For
a given $\Lambda$ let $\Lambda_{h}$ denotes the set of points at distance less
or equal to $h$ from the boundary of $\Lambda$ and let $|\Lambda_{h}|$ denotes
its volume. Let finally $d(\Lambda)$ denote the diameter of $\Lambda$ (i.e.
$d(\Lambda)~{}=~{}\sup_{x,y\in\Lambda}\\{|x-y|\\}$) .

Figure 6. A box $\Lambda$ going to infinity in a “non-thermodynamic” way

###### Definition 2.6

We say that $\Lambda$ tends to infinity in the sense of Fischer if
$|\Lambda|\to\infty$ and it exists a positive function $\pi(\alpha)$ such that
$\lim_{\alpha\to 0}\pi(\alpha)~{}=~{}0$ and for $\alpha$ sufficiently small
and for all $\Lambda$

${|\Lambda_{\alpha d(\Lambda)}|\over|\Lambda|}\leq\pi(\alpha)$

Figure 7. A set $\Lambda$ with $N_{+}~{}=~{}44$ and $N_{-}~{}=~{}20$

A rectangle $R$ of sizes $f(t),t$ such that $\lim_{t\to\infty}{f(t)\over
t}~{}=~{}0$ does not go to infinity in the sense of Fischer. As a matter of
fact

$d(R)~{}=~{}[t^{2}+f^{2}(t)]^{1/2}$ $|\Lambda_{\alpha
d(R)}|~{}=~{}2\alpha[t^{2}+f^{2}(t)]^{1/2}[t-2\alpha[t^{2}+f^{2}(t)]^{1/2}+f(t)]$
${|\Lambda_{\alpha
d(R)}|\over|R|}~{}=~{}{2\alpha[t^{2}+f^{2}(t)]^{1/2}[t-2\alpha[t^{2}+f^{2}(t)]^{1/2}+f(t)]\over
tf(t)}$ $~{}=~{}{2\alpha t^{2}\sqrt{1+{f^{2}(t)\over
t^{2}}}[1-2\alpha\sqrt{1+{f^{2}(t)\over t^{2}}}+{f(t)\over t}]\over tf(t)}$

For any fixed $\alpha$ the quantity above can be make large at will, as
$t\to\infty$. Thus it is not possible to find any $\pi(\alpha)$ such that
${V_{\alpha d(R)}(R)/V(R)}\leq\pi(\alpha)$ for all $R$ (i.e. for all $t$).

On the other hand, the square $S$ of size $(t,t)$ goes to infinity in the
sense of Fischer

$d(S)~{}=~{}\sqrt{2}t,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}|\Lambda_{\alpha
d(S)}|~{}=~{}4\alpha\sqrt{2}t[t-\alpha\sqrt{2}t]$ ${|\Lambda_{\alpha
d(S)}|\over|S|}~{}=~{}{4\alpha\sqrt{2}\,t[t-\alpha\sqrt{2}t]\over
t^{2}}~{}=~{}4\alpha\sqrt{2}[1-\alpha\sqrt{2}]\leq 4\alpha\sqrt{2}$

Hence one can choose $\pi(\alpha)~{}=~{}4\alpha\sqrt{2}$.

####  2.7 Example: finite range potentials

We will prove in this section the existence of the thermodynamic limit for the
function $\beta p(\beta,\lambda)$ defined in (2.33), but we will not treat the
general case, namely particles interacting via a pair potential stable and
tempered and $\Lambda$ going to infinity as e.g. Van Hove. We will rather give
our proof in a simpler case. Namely, we will put ourselves in $d=3$ dimensions