# Large Deviations for the Spatial and Energy Distributions of Systems of
Classical Ensemble

D. P. Shinde<EMAIL_ADDRESS>168-A, Panchsheel Colony, Pachgaon,
Kolhapur 416013, State-Maharashtra, India

###### Abstract

Boltzmann-Sanov and Cramer-Chernoff’s theorems provide large deviation
probabilities, entropy, and rate functions for the spatial distribution of
systems and the total internal energy of an ensemble respectively. By the
method of Lagrange’s undetermined multipliers, the results of both theorems,
the differentials of entropies or rate functions with respect to numbers of
systems, establish the statistical equilibrium condition. We connect the large
deviation statistics to the reversible and irreversible processes of the
spatial arrangements of the systems and the energy exchange of the ensemble
with the heat reservoir. We obtain the equalities between the differentials of
entropies and other thermodynamic variables in the reversible processes and
suggest inequalities of entropy productions for the irreversible processes.

A cluster of phase points in the phase space of a given dimension,
corresponding to various sets of the micro-states i.e. certain values of the
generalized coordinates and momenta of the member systems, represents the
micro-states of the whole ensemble [1, 2]. The macro-states and macroscopic
variables, the measurable quantities of the systems, and the ensemble are
defined for the various sets of the micro-states in their respective phase
spaces [1, 2]. The statistical entropy, a state function of macroscopic
variables and a measure of a multiplicity of micro-states for a given macro-
state, is calculated for the spatial distribution of the systems over the
states in the infinitesimal partitions of the phase space of an ensemble [3,
4]. These infinitesimal partitions are called the elements of extension-in-
phase or probability-region [5, 3, 4]. The Gibbs probability-coefficient
$P_{g}$, specified over either a multiple integral of micro-state variables or
a simple integral of internal energy, defines a probability of finding a
randomly selected system in the extension-in-phase [5, 3, 4]. This definition,
equivalently, is referred to as a ratio of the number of systems that fall
within the extension-in-phase to the total number $\mathbb{N}$ of the systems
[5]. For the systems of canonical and micro-canonical ensembles, $P_{g}$ or
the density-in-phase $\mathcal{D}=\mathbb{N}P_{g}$ expresses, respectively, in
terms of the exponential and delta functions of the internal energy [1, 2].
The canonical ensemble can be viewed as a finite or infinite set of micro-
canonical systems and vice versa [5]. The temperature, pressure, or other
external parameters determine the macro-states and alter the forms of $P_{g}$,
the internal energy, and entropy of the systems and the ensemble [1, 2].

Asymptotic mathematical methods [6, 7] are extremely useful for the
calculations of the probabilities of macroscopic variables and entropies of
the spatial arrangements of the systems and their energy exchange with the
surroundings [1, 2]. In the former case, the whole ensemble is isolated from
the surroundings whereas in the latter case, the ensemble is attached to the
heat reservoir [1, 2, 8]. The laws of large numbers, central limit theorem
(CLT), combinatorial counting method, the Stirling approximation, and the
asymptotic approximation of the complex integral determine the probability
distributions of the macroscopic variables, regarded as random variables, of
the classical and quantum statistical mechanical systems [1, 2, 6, 7]. The
probability distribution is related to the measure of the multiplicity of
micro-states, and in turn to the entropy of a macro-state [1, 2, 8]. Clausius
[9] introduced the concept of entropy in his mechanical formulation of the
second law of thermodynamics. For a statistical formulation of the second law,
Boltzmann [1, 2, 10] used the combinatorial counting method and the Stirling
approximation for the factorials of large numbers to find first, the arbitrary
spatial arrangements of systems and then, their most probable state with the
maximum entropy. Besides, he suggested the origins of entropy productions in
the irreversible processes [10]. The steepest descent or saddle point method
is used to calculate the mean value of the macroscopic variables using the
asymptotic approximation of a complex integral [1, 2, 11]. The methods of the
most probable and mean value show, asymptotically and as a consequence of
infinitely small fluctuations or deviations, identical results for the
statistical equilibrium situation [1, 2]. These methods explore the normal
deviations or fluctuations of the macroscopic random variables from their mean
values [1, 2].

The large deviation theory (LDT) [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25] deals with the asymptotic probabilities and their exponential
convergence rates, and the mathematical properties of the rate functions of
independent and dependent random variables beyond their normal deviations. The
abstract formalism of the probability theory generalizes the large deviation
bounds, properties of the rate function, and the form of the weak law of large
numbers and convergence in probability of the random variables [26, 27, 28,
29]. Moreover, in statistical mechanics [29, 30, 31, 32, 33, 34], the LDT
applied to the exclusion processes, many-particle systems, and chaotic
dynamics.

This letter focuses on the calculations of the probabilities, entropies, and
large deviation rate functions, using the simple mathematical forms of the
Boltzmann-Sanov [10, 35] and Cramer-Chernoff’s [36, 37] theorems,
respectively, for the spatial arrangements and a sum of internal energies of a
large number of closed systems of a classical ensemble. In addition, for both
the reversible and irreversible processes of the spatial arrangements and
energy exchange, the findings of both theorems enable us to obtain the
connections between the large deviation statistics and classical
thermodynamics.

We consider closed and isolated ensemble of the total volume $\mathbb{V}$
consisting of a large number $\mathbb{N}$ of macroscopic, distinguishable or
indistinguishable, weakly interacting systems that share total internal energy
$\mathbb{E}$ [1, 2]. No transport of matter takes place either in to or out of
the systems or the closed ensemble during any process under consideration. The
interaction energies between the systems are smaller than their internal
energies. Therefore we have assumed that the probabilities of random variables
of the quasi-independent systems do not affect each other [1, 2]. Following
the mathematical theory of probability [26, 27, 28, 29], the phase space
$\mathbb{R}$ of the ensemble is partitioned into $r$ partial regions or
partitions $R_{1},R_{2},..,R_{i},..,R_{r}$, where the subscript $i$ runs over
all partitions [1, 2, 3, 4]. Our first task is to calculate the joint
probability distribution of the spatial arrangement of systems
$(\\{N_{i}\\},(i=1,2,..,r))$ with internal energies
$(\\{E_{i}\\},(i=1,2,..,r))$ over the states of all $R_{i}$ partitions.
Statistically [38, 39], this can be viewed as a joint or multinomial
probability mass distribution of all the events, the sets or subsets of the
outcomes, of the generalized De Moivre-Bernoulli trials (See Supplementary
Material [40] (SM I)).

Let us see the approach in statistical mechanics for the spatial distribution
of the systems [1, 2]. Here we first construct a multinomial spatial
probability distribution and then calculate the Boltzmann-Sanov entropy and
large deviation rate functions. Many distinguishable ways of a realization of
the macroscopic configuration of the systems $(\\{N_{i}\\},(i=1,..,r))$
subject to two restrictive conditions are possible [1, 2]. These restrictive
conditions, respectively, fix the total number $\mathbb{N}$ of the systems and
total internal energy $\mathbb{E}$ of the ensemble, (i)
$\sum^{r}_{i=1}N_{i}=\mathbb{N}$ and (ii)
$\sum^{r}_{i=1}N_{i}E_{i}=\mathbb{E}=\mathbb{N}\overline{E}$, where $N_{i}\geq
0$, $E_{i}\geq 0$ [1, 2]. The ratio $\overline{E}=\mathbb{E}/\mathbb{N}$, the
mean energy per system, remains finite as $\mathbb{E}\to\infty$ and
$\mathbb{N}\to\infty$ [1, 2]. Since a system distributes randomly, and thus it
may lie in any one of the states of an $i^{th}$ partition with a probability
$G_{i}$. Then, the probability of exactly $N_{i}$ systems occupying the states
in the same partition is $G_{i}^{N_{i}}$ [1, 2]. The multinomial spatial
probability distribution $\mathbb{P}^{(b)}$ of the $\mathbb{N}$ quasi-
independent and distinguishable systems, satisfying the two restrictive
conditions, is obtained in such a way that exactly $N_{1}$ systems lie in
states of $R_{1}$ partition, $N_{2}$ in the $R_{2}$,.., and exactly $N_{r}$ in
the $R_{r}$. Using combinatorics, the probability distribution
$\mathbb{P}^{(b)}$ takes the final form [1, 2, 38, 39, 41, 42] (SM II.1)

$\mathbb{P}^{(b)}\left(\\{N_{i}\\};i=1,..,r\right)=\mathbb{N}!\prod^{r}_{i=1}\frac{G_{i}^{N_{i}}}{N_{i}!},$
(1)

where $G_{i}$’s are the relative volumes of the partitions, $0\leq G_{i}\leq
1$ $(i=1,2,..,r)$, $\sum^{r}_{i=1}G_{i}=1$, and
$\mathbb{N}!/(N_{1}!N_{2}!..N_{i}!...N_{r}!)$ is the multinomial coefficient
[38, 39, 41, 42, 43]. The natural logarithm on both sides of Eq. (1) gives

$\displaystyle\log\mathbb{P}^{(b)}\left(.\right)=\log\mathbb{N}!+\sum_{i=1}^{r}\left(N_{i}\log
G_{i}-\log N_{i}!\right).$ (2)
$\displaystyle\log\mathbb{P}^{(b)}\left(.\right)=\sum_{i=1}^{r}N_{i}\log\frac{\mathbb{N}G_{i}}{N_{i}}+\mathcal{O}{(\log\mathbb{N})}.$
(3)

Equation (3) follows from the use of Stirling’s approximation for the
factorials of large numbers $\mathbb{N}$ and $N_{i}$ at Eq. (2), $\log
N_{i}!=N_{i}\log N_{i}-N_{i}+\mathcal{O}(\log\mathbb{N}_{i})$, and
$\log\mathbb{N}!=\mathbb{N}\log\mathbb{N}-\mathbb{N}+\mathcal{O}(\log\mathbb{N})$
[1, 2, 29]. The right-hand side of Eq. (3) is the total Boltzmann-Sanov
entropy $\mathbb{S}^{(b)}$ of the ensemble [1, 2, 29], and thus we write
$k\log\mathbb{P}^{(b)}=\mathbb{S}^{(b)}+\mathcal{O}{(\log\mathbb{N})}$, where
$k$ is the Boltzmann constant. The entropy $\mathbb{S}^{(b)}$ is the sum of
the cross-entropy, $\mathbb{S}_{cross}=\sum_{i=1}^{r}N_{i}\log G_{i}$, and the
entropy of mixing,
$\mathbb{S}_{mix}=\sum_{i=1}^{r}N_{i}\log(\frac{\mathbb{N}}{N_{i}})$ [44, 45,
46]. To obtain the Boltzmann-Sanov entropy and rate functions per system [26,
27, 28, 29], divide both sides of Eq. (3) by $\mathbb{N}$, then we get

$\displaystyle\begin{split}&\frac{1}{\mathbb{N}}\log\mathbb{P}^{(b)}\left(\\{\frac{N_{i}}{\mathbb{N}}\\};i=1,..,r\right)=\sum_{i=1}^{r}F_{i}\log\frac{G_{i}}{F_{i}}+\tilde{\mathcal{O}}{(.)},\\\
&\frac{1}{\mathbb{N}}\log\mathbb{P}^{(b)}(.)=S^{(b)}_{1}/k+\tilde{\mathcal{O}}{(.)}=-I^{(b)}_{1}+\tilde{\mathcal{O}}{(.)},\end{split}$
(4)

where $F_{i}=\frac{N_{i}}{\mathbb{N}}$ and $\sum_{i=1}^{r}F_{i}=1$ [28, 29].
From the right-hand side of the upper line of Eq. (4), we have a non-negative
expression $\sum_{i=1}^{r}F_{i}\log(\frac{F_{i}}{G_{i}})\geq 0$; the equality
holds if and only if each $F_{i}$ equal to $G_{i}$ [28, 29]. The limit
$\lim\limits_{{\mathbb{N}}\to\infty}\log\mathbb{P}^{(b)}/\mathbb{N}$ on the
left-hand side of Eq. (4) exists, and as $\mathbb{N}\to\infty$, the term
$\tilde{\mathcal{O}}{(\frac{\log\mathbb{N}}{\mathbb{N}})}$ converges to zero
[28, 29]. On the right-hand side of Eq. (4),
$S^{(b)}_{1}=k\sum_{i=1}^{r}F_{i}\log(\frac{G_{i}}{F_{i}})$, and
$I^{(b)}_{1}=-\frac{S^{(b)}_{1}}{k}=\sum_{i=1}^{r}F_{i}\log(\frac{F_{i}}{G_{i}})$
are, respectively, the Boltzmann-Sanov entropy [47, 43] and rate function [26,
27, 28, 29] for a single system. In addition,
$\mathbb{S}^{(b)}=\mathbb{N}S^{(b)}_{1}$ and
$\mathbb{I}^{(b)}=\mathbb{N}I^{(b)}_{1}$ are corresponding functions for the
whole ensemble. Therefore, the multinomial spatial probability distribution in
Eq. (1) becomes, $\mathbb{P}^{(b)}\approx
e^{\frac{\mathbb{S}^{(b)}}{k}}=e^{-\mathbb{I}^{(b)}}=e^{-\mathbb{N}I^{(b)}_{1}}$.
Furthermore, the rate function $I^{(b)}_{1}$ is also known as Kullback–Leibler
(KL) divergence $D(F|G)$ between $F$ and $G$ [26, 27, 28, 29, 48, 49]. The
supplementary material [40] (SM II.2) provides the details of Eqs. (1), (2),
(3), and (4), and findings in the case of quasi-independent and
indistinguishable systems [1, 2, 50]. Also, we give results of the Boltzmann-
Sanov entropy and the large deviation rate function for the multinomial
spatial probability for the uniform distribution of quasi-independent and
distinguishable systems.

Now we consider the probability distribution of the total internal energy
$\mathbb{E}$ of the ensemble which exchanges only energy with the heat
reservoir of temperature $T$. Our aim is to calculate the Cramer-Chernoff
large deviation upper bound on $\mathbb{E}$ in terms of the generalized
Bienayme-Chebyshev-Markov (BCM) probability inequality [36, 37]. The real-
valued random variable $\mathbb{E}$ is non-negative ($\geq 0$) and it can be
written as either a sum of internal energies of weakly interacting individual
systems, $\mathbb{E}=\sum_{l=1}^{\mathbb{N}}\varepsilon_{l}$ or a sum over all
states of the partitions for which the systems have the same energies,
$\mathbb{E}=\sum_{i=1}^{r}N_{i}E_{i}$ [2, 28]. The CLT states that the sum
$\mathbb{E}$, under the generalized conditions, asymptotically, follows the
Laplace-Gauss distribution [1, 2, 38]. Moreover, the LDT deals with the
asymptotic probabilities and rate functions of each $\epsilon_{l}$ and the sum
$\mathbb{E}$ beyond their CLT scales [26, 27, 28, 29].

For $\mathbb{E}$, let us define a new random variable $e^{-\beta\mathbb{E}}$
with a real-valued parameter $\beta$ ($=1/kT$) [28, 29]. Through the moment
generating function (MGF), also known as a partition function, and its natural
logarithm, a cumulant generating function (CGF), we can calculate,
respectively, the moments and cumulants of $e^{-\beta\mathbb{E}}$ [28, 29].
Besides, the MGF or partition function $\mathbb{Z}^{(c)}(\beta)$ of the
ensemble is defined as the Laplace transform of a structure-function [7] or
the density of states $\Omega(\mathbb{E})$ [51],
$\mathbb{Z}^{(c)}(\beta)=\mathcal{E}(e^{-\beta\mathbb{E}})=\int^{\infty}_{0}e^{-\beta\mathbb{E}}\Omega(\mathbb{E})d\mathbb{E}$,
where the symbol ${\mathcal{E}}$ denotes mathematical expectation [28, 29].
For the multinomial distribution of the quasi-independent and distinguishable
[1, 2] or localized systems [51], the joint MGF or the partition function of
the ensemble is equal to the $\mathbb{N}$-th power of the partition function
$Z^{(c)}_{1}(\beta)$ of a single system
${\mathcal{E}}(e^{-\beta\sum_{i=1}^{r}N_{i}E_{i}})=\sum_{(\\{N_{i}\\};1\leq
i\leq r)}\prod^{r}_{i=1}\mathbb{N}!\frac{(G_{i}e^{-\beta
E_{i}})^{N_{i}}}{N_{i}!}=(\sum^{r}_{i=1}G_{i}e^{-\beta
E_{i}})^{\mathbb{N}}=(Z^{(c)}_{1}(\beta))^{\mathbb{N}}$ [1, 2] (SM III.1).
Here, the sum in the parentheses is taken for all possible values of $N_{i}$
subject to two restrictive conditions, and the last two expressions follow the
multinomial theorem [28].

In the case of the quasi-independent and distinguishable systems, the Cramer-
Chernoff large deviation upper bound $\mathbb{P}^{(c)}$ [26, 27, 28, 29, 36,
37] (SM III.2) on the probability inequality for $e^{-\beta\mathbb{E}}$ which
exceeds the given threshold is obtained as

$\mathbb{P}^{(c)}(e^{-\beta\mathbb{E}}\geq
e^{-\beta\mathbb{N}E})\leq\frac{{\mathcal{E}}(e^{-\beta\mathbb{E}})}{e^{-\beta\mathbb{N}E}}=e^{\beta\mathbb{N}E}(Z^{(c)}_{1}(\beta))^{\mathbb{N}},$
(5)

where the inequality holds for $\beta\geq 0$ and $E$ is the internal energy of
a system. The optimum value of $\beta$, chosen within the interval of
convergence of the joint MGF, provides the tightest optimum upper bound [26,
27, 28, 29]. To obtain the Gibbs canonical entropy [1, 2, 8] and Cramer-
Chernoff large deviation rate function [26, 27, 28, 29], first, take the
natural logarithm on both sides of Eq. (5) and then divide both sides by
$\mathbb{N}$. Finally, we have

$\displaystyle\log\mathbb{P}^{(c)}(.)$
$\displaystyle\leq\beta\mathbb{N}E+\mathbb{N}\log Z^{(c)}_{1}(\beta).$ (6)
$\displaystyle\frac{1}{\mathbb{N}}\log\mathbb{P}^{(c)}(.)$ $\displaystyle\leq
S^{(c)}_{1}(E)/k=-I^{(c)}_{1}(E).$ (7)

The limit
$\lim\limits_{{\mathbb{N}}\to\infty}\frac{\log\mathbb{P}^{(c)}}{\mathbb{N}}$
on the left-hand side of Eq. (7) exists [26, 27, 28, 29]. From the right-hand
sides of Eqs. (6) and (7), the functions
$\frac{S^{(c)}_{1}(E)}{k}=\min\limits_{\beta\geq 0}(\beta E+\log
Z^{(c)}_{1}(\beta))$ and
$I^{(c)}_{1}(E)=-\frac{S^{(c)}_{1}(E)}{k}=\max\limits_{\beta\geq 0}(-\beta
E-\log Z^{(c)}_{1}(\beta))$ are, respectively, the Gibbs canonical entropy [1,
2, 8] and Cramer-Chernoff’s rate function [26, 27, 28, 29] for a single
system. Moreover, for the whole ensemble, the respective functions are written
as $\mathbb{S}^{(c)}(E)=\mathbb{N}S^{(c)}_{1}(E)$ and
$\mathbb{I}^{(c)}(E)=\mathbb{N}I^{(c)}_{1}(E)$. Therefore, the probability
inequality in Eq. (5) takes the form $\mathbb{P}^{(c)}\leq
e^{\frac{\mathbb{S}^{(c)}}{k}}=e^{-\mathbb{I}^{(c)}}=e^{-\mathbb{N}I^{(c)}_{1}}$.
In general, both Gibbs canonical entropy and Cramer-Chernoff’s rate function
are the functions of the internal energy and external parameters, say,
$\lambda_{n}(n=1,2,..)$ [7, 51]. The minimum value of the parameter
$\beta=\beta^{*}$, holding the external parameters $\lambda_{n}$ constant,
gives the mean energy $\overline{E}=-(\partial\log
Z^{(c)}_{1}(\beta)/\partial\beta)_{\beta=\beta^{*},\lambda_{n}}$ [7, 51]. The
rate function $I^{(c)}_{1}$ attains the minimum at mean energy, i.e,
$I^{(c)}_{1}(\overline{E})=0$ [26, 27, 28, 29].

We find that the Cramer-Chernoff’s rate function is identical with the Gibbs
index of probability $\eta$ (the natural logarithm of Gibbs probability-
coefficient) [5], $I^{(c)}_{1}(E)=\eta(E)=\beta(A-E)$, where
$A(\beta)={-\beta}^{-1}\log Z^{(c)}_{1}(\beta)$ is the Helmholtz free energy
[1, 2, 8] or the work-function [51]. The Massieu potential [45] or Planck
characteristic function [51] per system is, $\psi_{m}(\beta)=-\beta A=\log
Z^{(c)}_{1}(\beta)$. The functions $I^{(c)}_{1}(E)$ and $\log
Z^{(c)}_{1}(\beta)$ or $\psi_{m}(\beta)$ show duality with each other via the
Legendre-Fenchel transform [26, 27, 28, 29], $\log
Z^{(c)}_{1}(\beta)=\psi_{m}(\beta)=\max\limits_{\beta}(-\beta
E-I^{(c)}_{1}(E))$, and $I^{(c)}_{1}(E)=\max\limits_{E}(-\beta
E-\psi_{m}(\beta))$.

For the quasi-independent and indistinguishable [1, 2] or non-localized
systems [51], we put the correct partition function of the ensemble
$\mathcal{E}(e^{-\beta\mathbb{E}})=\frac{Z^{(c)}_{1}(\beta))^{\mathbb{N}}}{\mathbb{N}!}$
[1, 2, 51] in Eq (5), then take the natural logarithm on both sides of
equation, and finally use the Stirling approximation for the factorial
$\log\mathbb{N}!=\mathbb{N}\log\mathbb{N}-\mathbb{N}+\mathcal{O}{(\log\mathbb{N})}$.
Putting the natural logarithm of the correct partition function
$\log(\frac{eZ^{(c)}_{1}(\beta)}{\mathbb{N}})$ and following Eqs (6) and Eq
(7), we get the expressions for the Gibbs canonical entropy [1, 2, 51],
Cramer-Chernoff’s rate function, and the Helmholtz free energy (For details,
see Supplementary Material [40] (SM III.2)).

Let us show that the results of both theorems, the entropies or logarithms of
the probability distributions, establish the statistical equilibrium
condition. The total Boltzmann-Sanov and Gibbs canonical entropies without any
additives, respectively, the right-hand sides of Eqs. (3) and (6), are the
functions of numbers $N_{i}$,
$\mathbb{S}^{(b)}=k\sum_{i=1}^{r}\Phi^{(b)}(N_{i})$ and
$\mathbb{S}^{(c)}=k\sum_{i=1}^{r}\Phi^{(c)}(N_{i})$, where
$\Phi^{(b)}(N_{i})=N_{i}\log(\frac{\mathbb{N}G_{i}}{N_{i}})$ and
$\Phi^{(c)}(N_{i})=\beta N_{i}E_{i}+N_{i}\log Z^{(c)}_{1}(\beta)$ [47, 43].
Both the most probable and mean values of $N_{i}$ are the indicators of
statistical equilibrium [47, 43]. The logarithms of probability distributions
$\log\mathbb{P}^{(b)}$ and $\log\mathbb{P}^{(c)}$, or the entropies
$\mathbb{S}^{(b)}$ and $\mathbb{S}^{(c)}$ reach their respective maximum
values at the statistical equilibrium [47, 43]. The total differentials
$d(\equiv\frac{\partial}{\partial N_{i}}dN_{i})$ of $\Phi^{(b)}(N_{i})$ and
$\Phi^{(c)}(N_{i})$ with respect to the numbers $N_{i}$, keeping other
parameters constant, are written as
$d(k\log\mathbb{P}^{(b)})=d\mathbb{S}^{(b)}\approx
k\sum_{i=1}^{r}\log(\frac{\mathbb{N}G_{i}}{N_{i}})dN_{i}$ and
$d(k\log\mathbb{P}^{(c)})=d\mathbb{S}^{(c)}\approx k\sum_{i=1}^{r}[\beta
E_{i}+\log Z^{(c)}_{1}(\beta)]dN_{i}$ respectively [47, 43]. In addition, the
differentials $d(\log\mathbb{P}^{(b)})$ and $d(\log\mathbb{P}^{(c)})$, subject
to the conditions $\sum_{i=1}^{r}dN_{i}=0$ and $\sum_{i=1}^{r}E_{i}dN_{i}=0$,
vanish at the equilibrium. Therefore, Lagrange’s method of undetermined
multipliers (SM IV.1) provides
$\sum_{i=1}^{r}[\log(\frac{\mathbb{N}G_{i}}{N_{i}})-\beta E_{i}-\log
Z^{(c)}_{1}(\beta)]dN_{i}=0$ [47, 43]. Thus, the equilibrium value $N_{i}^{*}$
of the numbers of systems is

$\frac{N_{i}^{*}}{\mathbb{N}}=\frac{G_{i}e^{-\beta
E_{i}}}{Z^{(c)}_{1}(\beta)}=G_{i}e^{\alpha-\beta E_{i}},$ (8)

where $e^{\alpha}=\frac{1}{Z^{(c)}_{1}(\beta)}=e^{\beta A}$ and the parameters
$\alpha$ and $\beta$ are the Lagrange’s undetermined multipliers [47, 43]. For
the equal $G_{i}$, Eq. (8) shows that the ratio $\frac{N_{i}^{*}}{\mathbb{N}}$
is equal to the probability $P(E_{i})=\frac{e^{-\beta
E_{i}}}{Z^{(c)}_{1}(\beta)}$ of finding a system in a state of energy $E_{i}$
[47, 43]. The substitution of $N_{i}^{*}$ from Eq. (8) into two restrictive
conditions gives, respectively, the canonical partition function and the mean
energy of a single system, $Z^{(c)}_{1}(\beta)=\sum_{i=1}^{r}G_{i}e^{-\beta
E_{i}}$ and $\overline{E}=\frac{\mathbb{E}}{\mathbb{N}}=-(\frac{\partial\log
Z^{(c)}_{1}(\beta)}{\partial\beta})_{\beta=\beta^{*}}$ [47, 43]. Putting this
$\overline{E}$ in Eq. (6), we get the expression for the mean canonical
entropy of a single system
$S^{(c)}_{1}(\overline{E})=-k\beta\frac{\partial\log
Z^{(c)}_{1}(\beta)}{\partial\beta}+k\log Z^{(c)}_{1}(\beta)$ [47, 43]. See
Supplementary Material [40] (SM IV.2) for the statistical equilibrium
condition, similar to Eq. (8), in the case of quasi-independent and
indistinguishable systems.

Using Eqs. (1) and (5) and our notations, for a coupled model of an ensemble
and a heat reservoir [51], we obtain the probability distribution of the
macro-state of the systems $(\\{N_{i}\\},(i=1,2,..,r))$ [51] and the relative
probability of occurrence of the fluctuations [52] where either
$\mathbb{S}^{(b)}=\mathbb{S}^{(c)}$ or $\mathbb{S}^{(b)}\neq\mathbb{S}^{(c)}$
(See Supplementary Material [40] (SM V)).

We now see the connection of the large deviation statistics with
thermodynamical reversible and irreversible processes via the differentials of
entropies or rate functions. Clausius [9] gave equality (inequality) between
the differential changes in the entropy and heat for the reversible
(irreversible) processes. Moreover, he also showed that the uncompensated
transformation [9] or the entropy production [53] is zero and strictly
positive ($>0$), respectively, in the reversible and irreversible processes.
From the classical thermodynamics [54, 55, 56, 57, 53], we know that
$d\mathbb{W}\geq\Delta\mathbb{A}$, $d\mathbb{W}_{rev}=\Delta\mathbb{A}$, and
$d\mathbb{W}_{irrev}>d\mathbb{W}_{rev}$, where $\mathbb{W}$’s and
$\Delta\mathbb{A}$ refer to work and the difference in Helmholtz free energy
respectively. Then, we have the criterion of irreversibility [53]
$d\mathbb{W}_{irrev}-d\mathbb{W}_{rev}=d\mathcal{L}>0$, where $d\mathcal{L}$
is the uncompensated transformation (dissipated heat or work). In the present
study, both the spatial arrangements of systems and the energy exchange
between the ensemble and the heat reservoir contribute to entropy production
[10]. In both processes, the uncompensated transformations are equal to the
differences in the differentials of the respective entropies and heat
$Td\mathbb{S}^{(b)}-d\mathbb{Q}^{(b)}=d\mathcal{L}^{(b)}\geq 0$ and
$Td\mathbb{S}^{(c)}-d\mathbb{Q}^{(c)}=d\mathcal{L}^{(c)}\geq 0$; the
equalities (inequalities) hold for the reversible (irreversible) processes [9,
53]. The $d\mathbb{Q}$’s and $d\mathbb{W}$’s are the non-exact differentials
[47, 43].

Consider the reversible and irreversible processes of the spatial distribution
of the systems. It is shown that the Kullback–Leibler (KL) divergence is equal
to $\beta$ times the mean of dissipated work $\overline{\mathbb{W}}_{diss}$,
[54, 55, 56, 57]. Besides, it is the difference between the irreversible and
reversible works,
$\overline{\mathbb{W}}_{diss}=\mathbb{W}_{irrev}-\mathbb{W}_{rev}\geq 0$ [54,
55, 56, 57]. Here, we assert that the Boltzmann-Sanov rate function
$\mathbb{I}^{(b)}$ or the KL divergence $D(F|G)$ between $F$ and $G$,
analogous to the findings of [54, 55, 56, 57], is equal to $\beta$ times the
mean of dissipated work,
$\mathbb{I}^{(b)}=\mathbb{N}\sum_{i=1}^{r}\frac{N_{i}}{\mathbb{N}}\log(\frac{N_{i}}{\mathbb{N}G_{i}})=\beta\overline{\mathbb{W}}^{(b)}_{diss}=\beta(\mathbb{W}^{(b)}_{irrev}-\mathbb{W}^{(b)}_{rev})$.
The $\mathbb{W}^{(b)}_{irrev}$ and $\mathbb{W}^{(b)}_{rev}$($=\Delta A^{(b)}$)
are derived in terms of the chemical potentials, respectively, from the cross-
entropy $\mathbb{S}_{cross}$ and the entropy of mixing $\mathbb{S}_{mix}$ [44,
45, 58, 59, 60, 61, 46, 62, 63]. Heat $\mathbb{Q}^{(b)}$ is produced during
the rearrangements of systems over the states in the partitions [43]. The
differential change in the Boltzmann-Sanov entropy is given as
$-\frac{d\mathbb{S}^{(b)}}{k}=d\mathbb{I}^{(b)}=\beta(d\mathbb{W}^{(b)}_{irrev}-\mathbb{W}^{(b)}_{rev})$.

In the reversible process of energy exchange between the ensemble and the heat
reservoir, following Eq. (6), the differential of total Gibbs canonical
entropy gives the equality,
$\frac{d\mathbb{S}^{(c)}}{k}=-d\mathbb{I}^{(c)}=\beta(d\overline{\mathbb{E}}-\overline{d\mathbb{W}}^{(c)})$,
where $d\overline{\mathbb{E}}$ and
$\overline{d\mathbb{W}}^{(c)}=-\beta^{-1}\sum_{n}(\frac{\partial\log\mathbb{Z}^{(c)}(\beta)}{\partial\lambda_{n}})d\lambda_{n}$
are the changes in the mean internal energy and mean work-done respectively
[47, 43]. Moreover, we know the relation between the heat change and entropy
in the reversible process, $\beta
d\mathbb{Q}^{(c)}=\frac{d\mathbb{S}^{(c)}}{k}$ [47, 43]. Furthermore, if we
set $d\mathbb{W}^{(c)}_{irrev}=-\mathbb{X}^{(c)}d\mathbb{V}$, and
$d\mathbb{W}^{(c)}_{rev}=d\mathbb{A}^{(c)}$ then
$d\mathbb{W}^{(c)}_{irrev}-d\mathbb{W}^{(c)}_{rev}=d\mathcal{L}^{(c)}\geq 0$,
where $\mathbb{X}^{(c)}$ and $\mathbb{V}$ are pressure and volume respectively
[53].

The total entropy of the ensemble can vary either by entropy production due to
the spatial arrangements of the systems in the interior of the ensemble, or
entropy exchange to or from the heat reservoir (exterior of the ensemble)
[53]. These two contributions, corresponding to the differentials of the total
Boltzmann-Sanov and Gibbs canonical entropies, are written as
$d\mathbb{S}^{(j)}=d\mathbb{S}^{(b)}+d\mathbb{S}^{(c)}\geq 0$, where the
equality $d\mathbb{S}^{(j)}=0$ gives the relation,
$-d\mathbb{S}^{(b)}=d\mathbb{S}^{(c)}$ for a reversible process and the
inequality ($>$ sign) indicates irreversibility. We suggest two inequalities
of the entropy productions, analogous to those mentioned in the classical
thermodynamics [53], respectively, in the processes of the spatial
arrangements of systems (matter) in the interior of the ensemble and the
energy exchange between the ensemble and the heat reservoir (external
surroundings), (i) $d\mathbb{S}^{(j)}-d\mathbb{S}^{(c)}=d\mathbb{S}^{(b)}\geq
0$, and (ii) $d\mathbb{S}^{(j)}-d\mathbb{S}^{(b)}=d\mathbb{S}^{(c)}\geq 0$,
where equal (greater than) sign refer to the reversible (irreversible)
processes.

Summary - This letter describes the applications of large deviation theorems
for the distribution of the numbers $N_{i}$ and sum $\mathbb{E}$ of internal
energies of the quasi-independent and closed classical systems. In both
distributions, the large deviation probabilities decay exponentially as the
ensemble size increases, i.e. as $\mathbb{N}\to\infty$. We find that the rate
functions, obtained from both theorems, are negative of their respective
entropies. For both distinguishable and indistinguishable quasi-independent
systems, we calculate only the large deviation upper bound on the probability
of total internal energy $\mathbb{E}$ of the ensemble. We identify that
Cramer-Chernoff’s rate function is identical with the Gibbs index of
probability and shows the duality with the Massieu potential or Planck’s
characteristic function via the Legendre-Fenchel transform. We have explored
the large deviation statistics to obtain the statistical equilibrium
condition, a probability distribution of the macro-state of the systems in a
coupled model of the ensemble and heat reservoir, and the equalities between
the differentials of entropies and other thermodynamical variables in the
reversible processes of the spatial arrangements of systems and energy
exchange between the ensemble and the heat reservoir. Also, we suggest
inequalities of the entropy productions for the irreversible processes. The
supplementary material [40] contains the necessary details of our results.

Supplementary Material : “Large Deviations for the Spatial and Energy
Distributions of Systems of a Classical Ensemble"

##  I (I) Statistics of the Multinomial Probability Distribution

The multinomial probability mass distribution is defined for all the events,
the sets or subsets, of the outcomes of the generalized De Moivre-Bernoulli
trials [38, 39]. For a single trial, “a single system can occupy any one of
the states of an $i$-th partition" is an $i$-th event with a probability
$G_{i}$. Therefore, in $\mathbb{N}$ independent trials, the event $i$ occurs
exactly $N_{i}$ times with the probability $G_{i}^{N_{i}}$ [38, 39]. The
multinomial probability distribution $\mathbb{P}^{(b)}$ of all the
conditionally independent events, subject to the restriction
$\sum^{r}_{i=1}N_{i}=\mathbb{N}$, is obtained in such a way that the first
event occurs exactly $N_{1}$ times, the second in $N_{2}$ times,…, and the
$r$-th event in exactly $N_{r}$ times [38, 39]. Therefore, $\mathbb{P}^{(b)}$
is a product of probabilities of the random variables $X_{i}$ which exactly
takes $N_{i}$ values in all events,
$\mathbb{P}^{(b)}\left(\\{N_{i}\\};i=1,..,r\right)=\mathbb{N}!\prod_{i=1}^{r}P(X_{i}=N_{i})$
[38, 39].

In a spatial distribution model [38, 39], the positions of systems in the
partitions are considered as random variables $X_{i}$. Let
$\mathcal{E}(X_{i})=\mathbb{N}G_{i}$ and
$\mathbb{D}(X_{i})=\sqrt{\mathbb{N}G_{i}(1-G_{i})}\approx\sqrt{\mathbb{N}}G_{i}$
are the means and standard deviations (s.d.) of the random variables $X_{i}$
respectively. The real-valued random variables $Y_{i}$’s in their standardized
forms $Y_{i}=(N_{i}-\mathbb{N}G_{i})/\sqrt{\mathbb{N}}G_{i}$ $(i=1,2,...,r)$
have zero means and unit standard deviations [38, 39]. The De Moivre-Laplace
theorem [38, 39, 36] states that the joint distribution function
$\mathcal{F}(Y_{i})$ $(i=1,2,...,r)$ of $Y_{i}$’s, independent of
$\mathbb{N}$, asymptotically, tends to the multivariate normal distribution
$\Phi(Y_{i})$. Moreover, the normal deviations of $Y_{i}$’s are of the order
of $\sqrt{\mathbb{N}}$. Furthermore, the joint moment generating function
(MGF) or characteristic function (CF) of $Y_{i}$’s $(i=1,2,...,r)$ leads,
asymptotically, to the MGF or CF of the multivariate normal distribution [38,
39]. The deviations of $Y_{i}$’s may vary beyond their normal distribution
scale i.e. of the order of $\mathbb{N}^{\zeta}$ $(0<\zeta<1)$ [36, 35]. The
large deviation theory (LDT) concerns the asymptotic behavior of the
probabilities of the random variables $Y_{i}$’s, $P(Y_{i}\geq
C\mathbb{N}^{\zeta})$, where $C>0$ or $P(Y_{i}\geq\gamma(\mathbb{N})$, where
$\gamma(\mathbb{N})=\mathcal{O}(\mathbb{N}^{1/2})$ [36, 35].

##  II (II) Large Deviations for the Multinomial Spatial Distribution

###  II.1 (i) Quasi-independent and Distinguishable systems

In the spatial arrangements of the quasi-independent and distinguishable
systems [10, 35, 1, 2], the multinomial spatial probability distribution
$\mathbb{P}^{(b)}$ is,

$\mathbb{P}^{(b)}\left(\\{N_{i}\\};i=1,..,r\right)=\mathbb{N}!\prod^{r}_{i=1}\frac{G_{i}^{N_{i}}}{N_{i}!},$
(9)

where $N_{i}\geq 0$, $\sum_{i=1}^{r}N_{i}=\mathbb{N}$, $0\leq G_{i}\leq 1$
$(i=1,2,..,r)$, and $\sum^{r}_{i=1}G_{i}=1$ [10, 35, 1, 2]. Taking the natural
logarithm on both sides of Eq. (9), we get

$\log\mathbb{P}^{(b)}\left(\\{N_{i}\\};i=1,..,r\right)=\log\mathbb{N}!+\sum_{i=1}^{r}\left(N_{i}\log
G_{i}-\log N_{i}!\right).\\\ $ (10)

Using Stirling’s approximation for the factorials of the large numbers
$\mathbb{N}$ and $N_{i}$,
$\mathbb{N}!=\mathbb{N}\log\mathbb{N}-\mathbb{N}+\mathcal{O}{(\log\mathbb{N})}$
and $N_{i}!=N_{i}\log N_{i}-N_{i}+\mathcal{O}{(\log N_{i})}$ [10, 35, 1, 2],
Eq. (10) becomes,

$\displaystyle\begin{split}\log\mathbb{P}^{(b)}\left(\\{N_{i}\\};i=1,..,r\right)&=\mathbb{N}\log\mathbb{N}-\mathbb{N}+\sum_{i=1}^{r}\left(N_{i}\log
G_{i}-N_{i}\log N_{i}+N_{i}\right)+\mathcal{O}{(\log\mathbb{N})},\\\
&=\left(\sum_{i=1}^{r}N_{i}\right)\log\mathbb{N}+\sum_{i=1}^{r}\left(N_{i}\log
G_{i}-N_{i}\log N_{i}\right)+\mathcal{O}{(\log\mathbb{N})},\\\
&=\sum_{i=1}^{r}\left(N_{i}\log\frac{\mathbb{N}G_{i}}{N_{i}}\right)+\mathcal{O}{(\log\mathbb{N})}.\end{split}$
(11)

The total Boltzmann-Sanov entropy is proportional to the natural logarithm of
the probability distribution, $\mathbb{S}^{(b)}=k\log\mathbb{P}^{(b)}$, where
$k$ is the Boltzmann constant. The division on both sides of the last line of
Eq. (11) by $\mathbb{N}$ gives the equations for the Boltzmann-Sanov entropy
and large deviation rate functions per system [26, 27, 28, 29, 37, 41, 42],

$\displaystyle\begin{split}\frac{1}{\mathbb{N}}\log\mathbb{P}^{(b)}\left(\\{\frac{N_{i}}{\mathbb{N}}\\};i=1,..,r\right)&=\sum_{i=1}^{r}\left(\frac{N_{i}}{\mathbb{N}}\log\frac{\mathbb{N}G_{i}}{N_{i}}\right)+\tilde{\mathcal{O}}{\left(\frac{\log\mathbb{N}}{\mathbb{N}}\right)},\\\
&=\sum_{i=1}^{r}\left(F_{i}\log\frac{G_{i}}{F_{i}}\right)+\tilde{\mathcal{O}}{\left(\frac{\log\mathbb{N}}{\mathbb{N}}\right)},\\\
&=S^{(b)}_{1}/k+\tilde{\mathcal{O}}{(.)}=-I^{(b)}_{1}+\tilde{\mathcal{O}}{(.)},\end{split}$
(12)

where $F_{i}=\frac{N_{i}}{N}$, and $\sum^{\mathbb{N}}_{i=1}F_{i}=1$ [26, 27,
28, 29, 37, 41, 42]. As $\mathbb{N}\to\infty$,
$\tilde{\mathcal{O}}{(\frac{\log\mathbb{N}}{\mathbb{N}})}\to 0$ [26, 27, 28,
29, 37, 41, 42]. For a single system,
$S^{(b)}_{1}=\frac{\mathbb{S}^{(b)}}{\mathbb{N}}\approx
k\sum_{i=1}^{r}F_{i}\log(\frac{G_{i}}{F_{i}})$, and
$I^{(b)}_{1}=\frac{\mathbb{I}^{(b)}}{\mathbb{N}}\approx-\frac{S^{(b)}_{1}}{k}=\sum_{i=1}^{r}F_{i}\log(\frac{F_{i}}{G_{i}})$
are, respectively, the Boltzmann-Sanov entropy and rate functions.

###  II.2 (ii) Uniform Multinomial Probability Distribution of Systems

For the multinomial spatial probability of the uniformly distributed systems
or the joint distribution of equally likely events with the probabilities,
$G_{i}=\frac{1}{r}$,$(i=1,2,..r)$ for each $i$, and following Eq. (9), (10),
(11), and (12), we obtain expressions, respectively, for the multinomial
probability $\mathbb{P}^{(b)}_{u}$ and its natural logarithm
$\log\mathbb{P}^{(b)}_{u}$, total Boltzmann-Sanov entropy
$\frac{\mathbb{S}^{(b)}_{u}}{k}$ and rate functions $\mathbb{I}^{(b)}_{u}$ of
the ensemble [27, 29, 4].

$\displaystyle\mathbb{P}^{(b)}_{u}\left(\\{N_{i}\\};i=1,..,r\right)=\mathbb{N}!\prod^{r}_{i=1}\frac{r^{-\mathbb{N}}}{N_{i}!}.$
(13)
$\displaystyle\log\mathbb{P}^{(b)}_{u}\left(\\{N_{i}\\};i=1,..,r\right)=\log\mathbb{N}!-\sum_{i=1}^{r}[\mathbb{N}\log
r-\log N_{i}!],$ (14)
$\displaystyle\log\mathbb{P}^{(b)}_{u}\left(\\{N_{i}\\};i=1,..,r\right)\approx-\mathbb{N}\log
r+\sum_{i=1}^{r}N_{i}\log\left(\frac{\mathbb{N}}{N_{i}}\right)=\frac{\mathbb{S}^{(b)}_{u}}{k}=-\mathbb{I}^{(b)}_{u}.$
$\displaystyle\frac{1}{\mathbb{N}}\log\mathbb{P}^{(b)}_{u}\left(\\{\frac{N_{i}}{\mathbb{N}}\\};i=1,..,r\right)\approx-\log
r+\sum_{i=1}^{r}F_{i}\log\left(\frac{1}{F_{i}}\right)=\frac{S^{(b)}_{1,u}}{k}=-I^{(b)}_{1,u}.$
(15)

where $\frac{S^{(b)}_{1,u}}{k}=-\log r-\sum_{i=1}^{r}F_{i}\log F_{i}$, and
$I^{(b)}_{1,u}=\log r+\sum_{i=1}^{r}F_{i}\log F_{i}$ are the entropy and rate
function for a single system respectively [27, 29, 4].

##  III (III) Large Deviations for the Total Internal Energy of the Ensemble

###  III.1 (i) Moment Generating Function or Partition Function of the
Ensemble

The generalized Bienayme-Chebyshev-Markov (BCM) probability inequality
expresses in terms of the moment generating function (MGF) and a given
threshold [38, 39, 28, 29]. For a multinomial distribution [47], the joint MGF
or partition function of the whole ensemble, subject to the constant total
number of systems and total internal energy $\mathbb{E}$ of the ensemble, is
written as

$\displaystyle{\mathcal{E}}(e^{-\beta\mathbb{E}})={\mathcal{E}}(e^{-\beta\sum_{i=1}^{r}N_{i}E_{i}})$
$\displaystyle=\prod^{r}_{i=1}\mathbb{N}!\frac{G_{i}^{N_{i}}}{N_{i}!}(e^{-\beta\sum_{i=1}^{r}N_{i}E_{i}})=\sum_{(\\{N_{i}\\};1\leq
i\leq r)}\prod^{r}_{i=1}\mathbb{N}!\frac{(G_{i}e^{-\beta
E_{i}})^{N_{i}}}{N_{i}!},$ (18) $\displaystyle=\sum_{(\\{N_{i}\\};1\leq i\leq
r)}\frac{\mathbb{N}!}{N_{1}!N_{2}!....N_{r}!}[(G_{1}e^{-\beta
E_{1}})^{N_{1}}*....*(G_{r}e^{-\beta E_{r}})^{N_{r}}],$
$\displaystyle=\left(\sum^{r}_{i=1}G_{i}e^{-\beta
E_{i}}\right)^{\mathbb{N}}=\left(Z^{(c)}_{1}(\beta)\right)^{\mathbb{N}},$

where the $\mathcal{E}$ symbol on the left-hand side of Eq. (18) denotes the
mathematical expectation [28, 29]. The right-hand side of Eq. (18) follows the
multinomial theorem [47]. The term $Z^{(c)}_{1}(\beta)$ is the partition
function of a single system. The MGF or partition function
$\mathbb{Z}^{(c)}(\beta)$ of the ensemble is also defined as the Laplace
transform of the density of states $\Omega(\mathbb{E})$[47, 51] or a
structure-function [7],
$\mathbb{Z}^{(c)}(\beta)=\mathcal{E}(e^{-\beta\mathbb{E}})=\int^{\infty}_{0}e^{-\beta\mathbb{E}}\Omega(\mathbb{E})d\mathbb{E}$.
For the quasi-independent systems and distinguishable systems, the partition
function of the ensemble is
$\mathbb{Z}^{(c)}(\beta)=\mathcal{E}(e^{-\beta\mathbb{E}})=\mathcal{E}(\prod^{\mathbb{N}}_{l=1}e^{-\beta\varepsilon_{l}})=\prod^{\mathbb{N}}_{l=1}\mathcal{E}(e^{-\beta\varepsilon_{l}})=[\mathcal{E}(e^{-\beta\varepsilon_{l}})]^{\mathbb{N}}=[Z^{(c)}_{1}(\beta)]^{\mathbb{N}}$,
where
$\mathcal{E}(e^{-\beta\varepsilon_{l}})=\int^{\infty}_{0}e^{-\beta\varepsilon_{l}}\omega(\varepsilon_{l})$
is the Laplace transform of the density of states $\omega(\varepsilon_{l})$ of
a single system [47, 7].

###  III.2 (ii) Cramer-Chernoff Large Deviation Upper Bound

We define a probability inequality [26, 27, 28, 29, 37] for the total internal
energy of the ensemble,
$\mathbb{P}^{(c)}(\mathbb{E}\leq\mathbb{N}E)=\mathbb{P}^{(c)}(-\mathbb{E}\geq-\mathbb{N}E)=\mathbb{P}^{(c)}(-\beta\mathbb{E}\geq-\beta\mathbb{N}E)=\mathbb{P}^{(c)}(e^{-\beta\mathbb{E}}\geq
e^{-\beta\mathbb{N}E})$, where $\beta(=\frac{1}{kT})\geq 0$, $T$ is the
absolute temperature, and $E$ is the energy of a single system.

(a) For the quasi-independent and distinguishable [47] or localized systems
[51], applying the Bienayme-Chebyshev-Markov inequality to the exponential
random variable $e^{-\beta\mathbb{E}}$, the Cramer-Chernoff large deviation
upper bound $\mathbb{P}^{(c)}$[26, 27, 28, 29, 37] is,

$\displaystyle\mathbb{P}^{(c)}(e^{-\beta\mathbb{E}}\geq
e^{-\beta\mathbb{N}E})$
$\displaystyle\leq\frac{{\mathcal{E}}(e^{-\beta\mathbb{E}})}{e^{-\beta\mathbb{N}E}}=\frac{{\mathcal{E}}(e^{-\beta\sum_{i=1}^{r}N_{i}E_{i}})}{e^{-\beta\mathbb{N}E}}=\frac{\prod^{r}_{i=1}{\mathcal{E}}(e^{-\beta
N_{i}E_{i}})}{e^{-\beta\mathbb{N}E}}=e^{\beta\mathbb{N}E}\left(Z^{(c)}_{1}(\beta)\right)^{\mathbb{N}},$
(19)

where we have put MGF from Eq. (18). Take the natural logarithm on both sides
of Eq. (19) and then divide both sides by $\mathbb{N}$,

$\displaystyle\log\mathbb{P}^{(c)}(e^{-\beta\mathbb{E}}\geq
e^{-\beta\mathbb{N}E})$ $\displaystyle\leq\mathbb{N}\left(\beta E+\log
Z^{(c)}_{1}(\beta)\right).$ (20)
$\displaystyle\frac{1}{\mathbb{N}}\log\mathbb{P}^{(c)}(e^{-\beta\mathbb{E}}\geq
e^{-\beta\mathbb{N}E})$ $\displaystyle\leq\beta E+\log Z^{(c)}_{1}(\beta).$
(21)

(b) In the case of quasi-independent and indistinguishable [1, 2, 47] or non-
localized systems [51], using the correct partition function
$\mathcal{E}(e^{-\beta\mathbb{E}})=\mathbb{Z}^{(c)}(\beta)=\frac{Z^{(c)}_{1}(\beta))^{\mathbb{N}}}{\mathbb{N}!}$
[1, 2, 47], and following the steps of Eqs. (19), (20), and (21), we get the
set of equations

$\displaystyle\mathcal{P}^{(c)}_{id}(e^{-\beta\mathbb{E}}\geq
e^{-\beta\mathbb{N}E})\leq\frac{(Z^{(c)}_{1}(\beta))^{\mathbb{N}}/\mathbb{N}!}{e^{-\beta\mathbb{N}E}}=e^{\beta\mathbb{N}E}\frac{(Z^{(c)}_{1}(\beta))^{\mathbb{N}}}{\mathbb{N}!}.$
(22) $\displaystyle\log\mathcal{P}^{(c)}_{id}(e^{-\beta\mathbb{E}}\geq
e^{-\beta\mathbb{N}E})\leq\mathbb{N}\beta E+\mathbb{N}\log
Z^{(c)}_{1}(\beta)-\log\mathbb{N}!.$ (23)
$\displaystyle\frac{1}{\mathbb{N}}\log\mathcal{P}^{(c)}_{id}(e^{-\beta\mathbb{E}}\geq
e^{-\beta\mathbb{N}E}\leq\beta
E+\log\left(\frac{eZ^{(c)}_{1}(\beta)}{\mathbb{N}}\right).$ (24)

We have used Stirling’s approximation for $\mathbb{N}!$ in Eq. (23).

(c) For the quasi-independent and distinguishable or localized systems, the
Gibbs canonical entropy $S^{(c)}_{1}(E)$ [5], Cramer-Chernoff rate function
$I^{(c)}_{1}(E)$ per system (See Eq. (21)) [28, 29] are, respectively, written
as $\frac{S^{(c)}_{1}(E)}{k}=\min_{\beta\geq 0}(\beta E+\log
Z^{(c)}_{1}(\beta))$ and $I^{(c)}_{1}(E)=\max_{\beta\geq 0}(-\beta E-\log
Z^{(c)}_{1}(\beta))=\max_{\beta\geq 0}(\beta(A-E))$, where
$A(\beta)={-\beta}^{-1}\log Z^{(c)}_{1}(\beta)$ is the Helmholtz free energy
or the work function of a single system [47, 51]. In the case of quasi-
independent and indistinguishable or non-localized systems, we put logarithm
of the correct partition function
$\log(\frac{eZ^{(c)}_{1}(\beta)}{\mathbb{N}})$ in the expressions of Gibbs
canonical entropy (See Eq. (24)), rate function, and Helmholtz free energy
[47, 51]. Moreover, $\mathbb{S}^{(c)}(E)=\mathbb{N}S^{(c)}_{1}(E)$ and
$\mathbb{I}^{(c)}(E)=\mathbb{N}I^{(c)}_{1}(E)$ are, respectively, the total
Gibbs canonical entropy and rate function of the ensemble.

We have found that the Cramer-Chernoff large deviation rate function is
identical with the Gibbs index of probability $\eta$, (the natural logarithm
of coefficient of probability $P_{g}$), $I^{(c)}_{1}(E)=\beta(A-E)=\eta=\log
P_{g}$ [5]. Thus we can write $P_{g}=e^{\eta}=e^{I^{(c)}_{1}(E)}$ [5].
Besides, the rate function, the negative of entropy, is also identical to the
Boltzmann $H$-function [47, 51].

##  IV (IV) Lagrange’s Method of Undetermined Multipliers

###  IV.1 (i) The Statistical Equilibrium Condition for Quasi-independent and
Distinguishable systems

We define two functions of numbers $N_{j}$, $\Phi^{(b)}(N_{j})$ and
$\Phi^{(c)}(N_{j})$ [47, 43] from Eqs. (11) and (20) respectively,

$\displaystyle\log\mathbb{P}^{(b)}\left(.\right)$
$\displaystyle=\sum_{j=1}^{r}\left(N_{j}\log\frac{\mathbb{N}G_{j}}{N_{j}}\right)+\mathcal{O}{(\log\mathbb{N})}=\sum_{j=1}^{r}\Phi^{(b)}(N_{j})+\mathcal{O}{(\log\mathbb{N})},$
(25) $\displaystyle\log\mathbb{P}^{(c)}(.)$
$\displaystyle=\sum_{j=1}^{r}\left(N_{j}\log Z^{(c)}_{1}(\beta)+\beta
N_{j}E_{j}\right)=\sum_{j=1}^{r}\Phi^{(c)}(N_{j}),$ (26)

where we have used $\mathbb{N}=\sum_{j=1}^{r}N_{j}$. From these Eqs. (25) and
(26), the respective total entropies are
$\mathbb{S}^{(b)}=k\log\mathbb{P}^{(b)}\left(.\right)\approx
k\sum_{j=1}^{r}N_{j}\log\frac{G_{j}}{N_{j}}+k\mathbb{N}\log\mathbb{N}$ and
$\mathbb{S}^{(c)}=k\log\mathbb{P}^{(c)}\left(.\right)=k\sum_{j=1}^{r}\left(N_{j}\log
Z^{(c)}_{1}(\beta)+\beta N_{j}E_{j}\right)$.

The total differentials $d\equiv(\frac{\partial}{\partial N_{i}}dN_{i})$ with
respect to the numbers $N_{j}$, for $j=i$, on both sides of Eqs (25) and (26)
provide, respectively,

$\displaystyle d\log\mathbb{P}^{(b)}\left(.\right)$
$\displaystyle=\sum_{i=1}^{r}\frac{\partial\Phi^{(b)}(N_{i})}{\partial
N_{i}}dN_{i}=\sum_{i=1}^{r}\left[\log\left(\frac{\mathbb{N}G_{i}}{N_{i}}\right)-\log
e\right]dN_{i},$ (27) $\displaystyle d\log\mathbb{P}^{(c)}(.)$
$\displaystyle=\sum_{i=1}^{r}\frac{\partial\Phi^{(c)}(N_{i})}{\partial
N_{i}}dN_{i}=\sum_{i=1}^{r}\left(\beta E_{i}+\log
Z^{(c)}_{1}(\beta)\right)dN_{i}.$ (28)

To obtain the statistical equilibrium condition, we use Lagrange’s method of
undetermined multipliers subject to conditions $\sum_{i=1}^{r}dN_{i}=0$ and
$\sum_{i=1}^{r}E_{i}dN_{i}=0$ [47, 43]. Using Eqs. (27) and (28), and ignoring
$\log e$, we get the equilibrium value of the numbers $N_{i}^{*}$,

$\displaystyle\sum_{i=1}^{r}\left[\log\left(\frac{\mathbb{N}G_{i}}{N_{i}}\right)-\beta
E_{i}-\log Z^{(c)}_{1}(\beta)\right]dN_{i}=0.$ (29)
$\displaystyle\frac{N_{i}^{*}}{\mathbb{N}}=\frac{G_{i}e^{-\beta
E_{i}}}{Z^{(c)}_{1}(\beta)}=G_{i}e^{\alpha-\beta E_{i}}.$ (30)

Here $e^{\alpha}=\frac{1}{Z^{(c)}_{1}(\beta)}=e^{\beta A}$, and the parameters
$\alpha$ and $\beta$ are Lagrange’s undetermined multipliers [47, 43].

###  IV.2 (ii) The Statistical Equilibrium Condition for Quasi-independent and
Indistinguishable systems

The division by $\mathbb{N}!$ on the right-hand side of Eq. (9) gives the
multinomial spatial probability distribution $\mathcal{P}^{(c)}_{id}(.)$ for
quasi-independent and indistinguishable systems [1, 2]. We obtain the
equations for the logarithm of probability distribution and total Boltzmann-
Sanov entropy respectively.

$\displaystyle\mathcal{P}^{(b)}_{id}\left(\\{N_{i}\\};i=1,..,r\right)=\prod^{r}_{i=1}\frac{G_{i}^{N_{i}}}{N_{i}!}$
(31)
$\displaystyle\log\mathcal{P}^{(b)}_{id}(.)\approx\sum_{i=1}^{r}\left(N_{i}\log\frac{G_{i}}{N_{i}}+N_{i}\right).$
(32)
$\displaystyle\mathcal{S}^{(b)}_{id}=k\log\mathcal{P}^{(b)}_{id}(.)\approx
k\sum_{i=1}^{r}\left(N_{i}\log\frac{G_{i}}{N_{i}}\right)+k\mathbb{N}.$ (33)

From Eq. (22), we get the logarithm of Cramer-Chernoff probability
distribution and the Gibbs canonical entropy

$\displaystyle\log\mathcal{P}^{(c)}_{id}(.)$
$\displaystyle\approx\sum_{i=1}^{r}N_{i}\left(\log Z^{(c)}_{1}(\beta)+\beta
E_{i}-\log\mathbb{N}+\log e\right).$ (34)
$\displaystyle\mathcal{S}^{(c)}_{id}=k\log\mathcal{P}^{(c)}_{id}(.)$
$\displaystyle\approx k\sum_{i=1}^{r}\left(N_{i}\log Z^{(c)}_{1}(\beta)+\beta
N_{i}E_{i}\right)+k\mathbb{N}-k\mathbb{N}\log\mathbb{N}.$ (35)

Following Lagrange’s undetermined multipliers method, we finally obtain

$\frac{N_{i}^{*}}{\mathbb{N}}=\frac{G_{i}e^{-\beta
E_{i}}}{Z^{(c)}_{1}(\beta)}.$ (36)

Equations (30) and (36) are identical. The differences between the entropies
of indistinguishable and distinguishable cases of the systems are written as
$\mathcal{S}^{(b)}_{id}-\mathbb{S}^{(b)}\approx
k(\mathbb{N}-\mathbb{N}\log\mathbb{N})$ and
$\mathcal{S}^{(c)}_{id}-\mathbb{S}^{(c)}\approx
k(\mathbb{N}-\mathbb{N}\log\mathbb{N})$.

Alternatively, we can obtain the equilibrium conditions (30) and (36), by the
same method of Lagrange, using the variations of the rate functions
$I^{(b)}_{1}$ and $I^{(c)}_{1}$ with respect to
$F_{i}(=\frac{N_{i}}{\mathbb{N}})$ respectively.

##  V (V) Coupled Model of Ensemble and Heat Reservoir

Fowler [51] obtained a relation between probability and entropy in the case of
a coupled model. In this model, the ensemble is immersed in a heat reservoir
which has a very large number of localized systems than the ensemble. Using
Eqs. (9) and (19), the probability distribution $\mathbb{P}^{(f)}$ [51] of the
state of systems in the coupled model is,

$\mathbb{P}^{(f)}=\frac{\mathbb{P}^{(b)}}{\mathbb{P}^{(c)}}=\left(\mathbb{N}!\prod^{r}_{i=1}\frac{G_{i}^{N_{i}}}{N_{i}!}\right).\left[\frac{e^{-\beta\mathbb{N}E}}{Z^{(c)}_{1}(\beta)^{\mathbb{N}}}\right].$
(37)

The natural logarithm on both sides of Eq. (37) supplies

$\log\mathbb{P}^{(f)}=\log\mathbb{N}!+\sum_{i=1}^{r}\left(N_{i}\log G_{i}-\log
N_{i}!\right)-\left[\beta\mathbb{N}E+\mathbb{N}\log
Z^{(c)}_{1}(\beta)\right].$ (38)

We can write the ratio of probabilities in Eq. (37) as

$\displaystyle\begin{split}&\mathbb{P}^{(f)}=\frac{\mathbb{P}^{(b)}}{\mathbb{P}_{max}^{(c)}}=e^{(\mathbb{S}^{(b)}-\mathbb{S}^{(c)}_{max})/k},\\\
&\log\mathbb{P}^{(f)}=\log\left(\frac{\mathbb{P}^{(b)}}{\mathbb{P}_{max}^{(c)}}\right)=\frac{\mathbb{S}^{(b)}-\mathbb{S}^{(c)}_{max}}{k},\end{split}$
(39)

where
$\mathbb{S}^{(b)}=\mathbb{N}k\sum_{i=1}^{r}N_{i}\log(\frac{\mathbb{N}G_{i}}{N_{i}})$
and $\mathbb{S}^{(c)}_{max}=\mathbb{N}\beta\overline{E}+\mathbb{N}\log
Z^{(c)}_{1}(\beta)$, respectively, the total Boltzmann-Sanov and Gibbs
canonical entropies. In Eq. (39), $\mathbb{P}_{max}^{(c)}$ and
$\mathbb{S}^{(c)}_{max})/k$ are the maximums, respectively, of Cramer-
Chernoff’s probability and Gibbs canonical entropy. Moreover, the Eq. (39) can
be seen as the relative probability of the occurrence of the fluctuations [1,
2, 52] where both these entropies may or may not be equal.

For the quasi-independent and indistinguishable case, we have the similar
expression
$\mathcal{P}^{(f)}_{id}=\frac{\mathcal{P}^{(b)}_{id}}{\mathcal{P}^{(c)}_{id}}=\left(\prod^{r}_{i=1}\frac{G_{i}^{N_{i}}}{N_{i}!}\right).\left[\frac{\mathbb{N}!e^{-\beta\mathbb{N}\overline{E}}}{Z^{(c)}_{1}(\beta)^{\mathbb{N}}}\right]$.

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# Condensed Ricci Curvature of Complete and Strongly Regular Graphs

Vincent Bonini  ,  Conor Carroll  ,  Uyen Dinh  ,  Sydney Dye  ,
Joshua Frederick  and  Erin Pearse

###### Abstract.

We study a modified notion of Ollivier’s coarse Ricci curvature on graphs
introduced by Lin, Lu, and Yau in [11]. We establish a rigidity theorem for
complete graphs that shows a connected finite simple graph is complete if and
only if the Ricci curvature is strictly greater than one. We then derive
explicit Ricci curvature formulas for strongly regular graphs in terms of the
graph parameters and the size of a maximal matching in the core neighborhood.
As a consequence we are able to derive exact Ricci curvature formulas for
strongly regular graphs of girth 4 and 5 using elementary means. An example is
provided that shows there is no exact formula for the Ricci curvature for
strongly regular graphs of girth $3$ that is purely in terms of graph
parameters.

##  1\. Introduction

Lott, Villani [13] and Sturm [18] discovered a relationship between optimal
transport and Ricci curvature on smooth Riemannian manifolds and they pursued
a construction of a synthetic notion of Ricci curvature that could be defined
independently of differentiable structures. Ollivier [15, 16] later introduced
a notion of coarse Ricci curvature for Markov chains on metric spaces which
has a particularly accessible formulation on graphs. Much work has been done
on coarse Ricci curvature on graphs (cf. [1, 11, 12, 9, 17, 3, 4, 14]).

Ollivier’s coarse Ricci curvature on graphs is defined in terms of transport
distance of probability measures (see Section 2). We study a modified notion
of Ollivier’s coarse Ricci curvature on graphs introduced by Lin, Lu, and Yau
in [11]. In this paper we refer to the modified Ricci curvature of [11] as the
condensed Ricci curvature. Our first result concerning the condensed Ricci
curvature exploits the relationship between Ricci curvature and the
eigenvalues of the spectral graph Laplacian to establish a rigidity theorem
for complete graphs. This result is stated without proof in [11, Ex. 1].

######  Theorem 1.1.

A connected finite simple graph $G=(V,E)$ is complete if and only if the
condensed Ricci curvature $\Bbbk(x,y)>1$ for all edges $xy\in E$. In this
case, $\Bbbk(x,y)=\frac{n}{n-1}$ for all vertices $x,y\in V$.

We then turn our attention to the derivation of exact formulas for the
condensed Ricci curvature on strongly regular graphs. Explicit formulas and
curvature bounds for various forms of Ricci curvature have been established
for large classes of graphs (cf. [11, 12, 9, 17, 3, 7]). In particular, Bakry-
Émery Ricci curvature functions are studied in [7] and explicit formulas for
the Bakry-Émery Ricci curvature are derived for Cayley graphs and strongly
regular graphs of girths $4$ and $5$. This work served as our inspiration to
establish explicit formulas for the condensed Ricci curvature on strongly
regular graphs using elementary means.

Let $G=(V,E)$ be a strongly regular graph with parameters
$(n,d,\alpha,\beta)$. Here $n=|V|$ is the number of vertices, $d$ is the
uniform degree of the vertices, $\alpha\geq 0$ is the number of common
neighbors for adjacent vertices, and $\beta\geq 1$ is the number of common
neighbors for nonadjacent vertices. Given an edge $xy\in E$, let $N_{x}$
denote the set of vertices that are adjacent to $x$, not including $y$ or any
vertices that are adjacent to $y$. Similarly, let $N_{y}$ denote the set of
vertices that are adjacent to $y$, not including $x$ or any vertices that are
adjacent to $x$ (see Section 2 for more details). We compute the following
curvature formulas solely from properties of maximum matchings and the
regularity properties of strongly regular graphs.

######  Theorem 1.2.

Let $G=(V,E)$ be strongly regular graph with parameters $(n,d,\alpha,\beta)$.
Suppose $xy\in E$ with maximum matching $\mathcal{M}$ of size $m$ between
$N_{x}$ and $N_{y}$. Then the condensed Ricci curvature

(1.1) $\Bbbk(x,y)=\frac{\alpha+2}{d}-\frac{|N_{x}|-m}{d}.$

######  Remark 1.3.

We would like to point out that Theorem 1.2 holds more generally for regular
graphs of diameter $2$. In this case one takes $\alpha$ to be the number of
neighbors common to both $x$ and $y$. For simplicity in our presentation, we
focus on strongly regular graphs.

The explicit formula $\Bbbk(x,y)=\frac{\alpha+2}{d}$ was previously
established in [17, Theorem 6.3] for general graphs with perfect matchings
between $N_{x}$ and $N_{y}$. In the setting of strongly regular graphs Theorem
1.2 is thus an extension of the results of [17] to the case of maximum
matchings. These results give insight into the importance of understanding
matchings between the neighbor sets $N_{x}$ and $N_{y}$ in the derivation of
exact Ricci curvature formulas. In particular, we are able to exploit this
idea to derive the explicit formulas for the Ricci curvature of strongly
regular graphs of girths $4$ and $5$ using elementary means.

For strongly regular graphs $G=(V,E)$ of girth $5$, it follows that adjacent
vertices share $\alpha=0$ common neighbors. Moreover, for a given edge $xy\in
E$, there are no edges between the neighbor sets $N_{x}$ and $N_{y}$. Then, as
a simple consequence of Theorem 1.2, we have the following Ricci curvature
formula.

######  Theorem 1.4.

Let $G=(V,E)$ be strongly regular graph with parameters $(n,d,\alpha,\beta)$.
If the girth of $G$ is $5$, then condensed Ricci curvature

$\Bbbk(x,y)=\frac{3}{d}-1$

for all $xy\in E$.

From Theorem 1.4 it is easy to see that the condensed Ricci curvature along
any edge of the $5$-cycle $C_{5}$ is $\Bbbk=\frac{1}{2}$ while the condensed
Ricci curvature along any edge of the Peterson graph is $\Bbbk=0$. The fact
that these are the only nonnegatively curved strongly regular graphs of girth
$5$ then follows from Theorem 1.4 and the classification of Moore graphs of
diameter $2$ due to Hoffman and Singleton [8].

######  Corollary 1.5.

The $5$-cycle and the Petersen graph are the only strongly regular graphs of
girth $5$ with nonnegative condensed Ricci curvature along edges.

For strongly regular graphs $G=(V,E)$ of girth $4$, it follows that adjacent
vertices share $\alpha=0$ common neighbors. For a given edge $xy\in E$, we
appeal to Hall’s theorem and the pigeonhole principle to establish a perfect
matching between the neighbor sets $N_{x}$ and $N_{y}$. Then, as a consequence
of Theorem 1.2, we have the following Ricci curvature formula.

######  Theorem 1.6.

Let $G=(V,E)$ be strongly regular graph with parameters $(n,d,\alpha,\beta)$.
If the girth of $G$ is $4$, then condensed Ricci curvature

$\Bbbk(x,y)=\frac{2}{d}$

for all $xy\in E$. In particular, all strongly regular graphs of girth $4$
have positive condensed Ricci curvature.

For strongly regular graphs with girth $3$, it turns out that there does not
exist an exact formula for the condensed Ricci curvature solely in terms of
the graph parameters. This is due to the fact that there are nonisomorphic
girth $3$ strongly regular graphs with the same graph parameters but different
Ricci curvatures. For example, the $4\times 4$ Rook’s graph and the Shrikhande
graph both have parameters $(16,6,2,2)$ but their condensed Ricci curvatures
are $\Bbbk=\frac{2}{3}$ and $\Bbbk=\frac{1}{3}$, respectively. This is due to
the fact that for a given edge $xy$ there is a perfect matching between
neighbor sets $N_{x}$ and $N_{y}$ for the Rook’s graph but the maximum
matching between $N_{x}$ and $N_{y}$ is size $m=1$ for the Shrikhande graph.

On the other hand, strongly regular conference graphs exhibit many symmetry
properties that we believe always lead to perfect matchings between neighbor
sets $N_{x}$ and $N_{y}$. We plan to address conference graphs in a
forthcoming paper. For now we make the following conjecture for conference
graphs, which can be realized as a direct consequence of Theorem 1.2 provided
there are always perfect matchings between $N_{x}$ and $N_{y}$.

######  Conjecture 1.7.

Let $G=(V,E)$ be a strongly regular conference graph with parameters
$(4\beta+1,2\beta,\beta-1,\beta)$ with $\beta\geq 2$. Then the condensed Ricci
curvature

$\Bbbk(x,y)=\frac{1}{2}+\frac{1}{2\beta}$

for all $xy\in E$. In particular, all strongly regular conference graphs have
positive condensed Ricci curvature strictly greater than $\frac{1}{2}$.

This paper is organized as follows: In Section 2 we formally define transport
distance between probability measures on graphs and the condensed Ricci
curvature. We briefly discuss our choice of notation and terminology and we
state some key results from [4, 3] that simplify the computation of condensed
Ricci curvature. In Section 3 we introduce the spectral graph Laplacian and
prove Theorem 1.1. In Section 4 we focus on strongly regular graphs and
establish our exact curvature formulas.

## Acknowledgements

The authors were partially supported by NSF FURST grant DMS-$1620552$, the Cal
Poly Frost Fund, CSU Fresno, and Cal Poly, San Luis Obispo. The authors would
like to thank CSU Fresno for their hospitality during their summer
participation in the FURST program.

##  2\. Transport Distance and Condensed Ricci Curvature on Graphs

To motivate Ollivier’s formulation of coarse Ricci curvature we turn to the
setting of smooth Riemannian manifolds. Suppose that $x_{1}$ and $x_{2}$ are
points on an $N$-dimensional Riemannian manifold $M$ and that $m_{i}$ is the
uniform measure on the respective geodesic ball $B(x_{i},\varepsilon)$ of
radius $\varepsilon$ centered at $x_{i}$. Let $\delta=d(x_{1},x_{2})$ denote
the geodesic distance between $x_{1}$ and $x_{2}$. If $B(x_{1},\varepsilon)$
is parallel transported to $B(x_{2},\varepsilon)$ along a geodesic from
$x_{1}$ to $x_{2}$ in the direction of tangent vector $v$ at $x_{1}$, then the
transport distance $W(m_{1},m_{2})$ between the measures and the Ricci
curvature on $M$ satisfy the relation [15]

$\displaystyle
W(m_{1},m_{2})=\delta\left(1-\frac{\varepsilon^{2}}{2(N+2)}\operatorname{Ric}(v,v)+O(\varepsilon^{3}+\varepsilon^{2}\delta)\right)\quad\text{as}\
\varepsilon,\delta\to 0.$

As noted in [16], this relation allows one to interpret Ricci curvature on
Riemannian manifolds as a measure of the average distance traveled between
points of geodesic balls versus their centers under parallel transport. Using
this relationship to guide intuition, Ollivier defined the Ricci curvature of
Markov chains on metric spaces

$\displaystyle\operatorname{Ric}(x_{1},x_{2})=1-\frac{W(m_{1},m_{2})}{d(x_{1},x_{2})}$

where $W(m_{1},m_{2})$ denotes the general transport or Wasserstein distance
between measures $m_{i}$ based at $x_{i}$ and $d(x_{1},x_{2})$ is the metric
distance between $x_{1}$ and $x_{2}$. This definition provides a synthetic
notion of Ricci curvature of Markov chains on metric spaces and its geometric
validation comes in the form of analogues of the classical theorems of Bonnet-
Myers, Lichnerowicz, and Gromov-Lévy [15, 16].

Ollivier’s coarse Ricci curvature on graphs is defined in terms of transport
distance of probability measures and standard graph distance [15, 16]. Let
$G=(V,E)$ be a nontrivial, locally finite, undirected, connected, simple graph
with standard graph shortest path distance function

$\rho:V\times V\to{\mathbb{N}}\cup\\{0\\}.$

A probability measure or probability mass distribution on $G=(V,E)$ is a real
valued function $\mu:V\to[0,1]$ such that

$\sum_{v\in V}\mu(v)=1.$

Given probability measures $\mu$ and $\nu$ on $G$, a coupling or transport
plan between $\mu$ and $\nu$ is a probability measure $\xi:V\times V\to[0,1]$
such that

$\sum_{w\in V}\xi(v,w)=\mu\quad\text{and}\quad\sum_{v\in V}\xi(v,w)=\nu.$

The terminology transport plan has economic origins and comes from the optimal
transport problem where one interprets a coupling as a plan for transporting
mass or goods from one distribution center to another.

The $L^{1}$-Wasserstein or transport distance is a metric on the space of
probability measures on $G$ that quantifies the optimal transport cost between
probability measures on $G$.

######  Definition 2.1.

The $L^{1}$-Wasserstein or transport distance between probability measures
$\mu$ and $\nu$ on a graph $G=(V,E)$ is given by

$W(\mu,\nu)=\inf_{\xi\in{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}(\mu,\nu)}\sum_{x\in
V}\sum_{y\in V}\rho(x,y)\xi(x,y)$

where
${{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}(\mu,\nu)$
is the set of all transport plans between $\mu$ and $\nu$.

We consider a modified notion of Ollivier’s coarse Ricci curvature on graphs
introduced by Lin, Lu, and Yau in [11] that is based on probability measures
of the form

(2.1) $m_{x}^{\varepsilon}(y)=\begin{cases}1-\varepsilon&y=x,\\\
\frac{\varepsilon}{\deg(x)}&y\in\Gamma(x),\\\ 0&\text{otherwise}.\end{cases}$

For $x,y\in V$ the $\varepsilon$-Ollivier Ricci curvature is then defined for
$0\leq\varepsilon\leq 1$ by

$\kappa_{\varepsilon}(x,y)=1-\frac{W(m_{x}^{\varepsilon},m_{y}^{\varepsilon})}{\rho(x,y)}$

and the modified Ollivier Ricci curvature of is then defined for $x,y\in V$ by

$\Bbbk(x,y)=\lim_{\varepsilon\to
0}\frac{\kappa_{\varepsilon}(x,y)}{\varepsilon}.$

We refer to the modified Ricci curvature as the condensed Ricci curvature. Our
definitions differ slightly from those of [11] in that we use $1-\varepsilon$
in place of $\alpha$ as in [17, 14]. The reason for this choice in notation
and terminology is that we interpret the probability measure (2.1) as encoding
an $\varepsilon$-active random walk that becomes stationary as $\varepsilon\to
0$. One can view this phenomena as a concentration or condensation of measure.
This formulation also more closely resembles the notion of coarse Ricci
curvature of continuous-time Markov chains given by the derivative

$\kappa(x,y)=-\frac{d}{dt}\frac{W(m_{x}^{t},m_{y}^{t})}{\rho(x,y)}.$

Computing Ricci curvature on graphs requires solving the optimal transport
problem. By definition a given transport plan between measures
$m_{x}^{\varepsilon}$ and $m_{x}^{\varepsilon}$ gives an upper bound for the
transport distance $W(m_{x}^{\varepsilon},m_{y}^{\varepsilon})$ and therefore
a lower bound for condensed Ricci curvature. While an optimal transport plan
may be intuitive, realizing that it achieves the transport distance directly
from the definition is technically challenging as it requires solving a linear
programming problem. Fortunately, there is a dual formulation of this
optimization problem in terms of $1$-Lipschitz functions that one can use to
provide lower bounds to transport distance and consequently upper bounds for
condensed Ricci curvature.

######  Theorem 2.2 (Kantorovich Duality Theorem).

The $L^{1}$-Wasserstein or transport distance between probability measures
$\mu$ and $\nu$ on a graph $G=(V,E)$ is given by

$W(\mu,\nu)=\sup_{f\in Lip(1)}\sum_{z\in V}f(z)(\mu(z)-\nu(z))$

where

$Lip(1)=\\{f:V\to\mathbb{R}\ \big{|}\ \lvert f(x)-f(y)\rvert\leq\rho(x,y)\
\text{for all}\ x,y\in V\\}$

is the space of $1$-Lipschitz functions on $G$.

Due to the Kantorovich Duality Theorem we refer to $1$-Lipschitz functions on
a graph $G=(V,E)$ as Kantorovich potential functions. With the Kantorovich
Duality Theorem in hand one can compute exact formulas for Ricci curvature by
specifying a transport plan and a Kantorovich potential that provide bounds
that pinch the condensed Ricci curvature and force equality. Another useful
fact for computing condensed Ricci curvature due to [4] is that the Ricci
activeness function $\varepsilon\mapsto\Bbbk_{\varepsilon}$ is piecewise
linear with $2$ linear parts on regular graphs of degree $d$ and the
$\varepsilon$-Ollivier Ricci curvature differs from the condensed Ricci
curvature only by scale for all values $0\leq\varepsilon\leq\frac{d}{d+1}$.
More precisely, on a regular graph $G=(V,E)$ of degree $d$, for any $x,y\in V$

$\kappa_{\varepsilon}(x,y)=\varepsilon\ \Bbbk(x,y)\quad\text{for}\
0\leq\varepsilon\leq\frac{d}{d+1}.$

For simplicity, in our computations we take $\varepsilon=\frac{1}{2}$ and
compute the condensed Ricci curvature

$\Bbbk(x,y)=2\kappa_{\frac{1}{2}}(x,y).$

In Riemannian geometry, Ricci curvature is a local quantity so in the context
of graphs it seems natural to restrict one’s attention to computing Ricci
curvature along edges. With our conventions, this restriction allows for
further simplification in computing Ricci curvature due to [3]. For vertices
$x\in V$ let

$\Gamma(x)=\\{y\in V\ \big{|}\ \rho(x,y)=1\\}=\\{y\in V\ \big{|}\ xy\in E\\}$

denote the neighbor set of $x$. Given an edge $xy\in E$ we denote the set of
common neighbors or triangle set of $x$ and $y$ by

$\nabla_{xy}=\Gamma(x)\cap\Gamma(y).$

Then we further decompose the $1$-step neighborhoods of $x$ and $y$ into
disjoint unions

$\Gamma(x)=N_{x}\cup\nabla_{xy}\cup\\{y\\}\quad\text{and}\quad\Gamma(y)=N_{y}\cup\nabla_{xy}\cup\\{x\\}$

where

$N_{x}=\Gamma(x)\setminus(\nabla_{xy}\cup\\{y\\})\quad\text{and}\quad
N_{y}=\Gamma(y)\setminus(\nabla_{xy}\cup\\{x\\}).$

We denote the set of common $2$-step neighbors or pentagon set of $x$ and $y$
by

$P_{xy}=\\{z\in V\ \big{|}\ \rho(x,z)=2\ \text{ and }\ \rho(y,z)=2\\}.$

As in [3], we refer to the disjoint union of vertices

(2.2) $\mathcal{N}_{xy}=\\{x\\}\cup\\{y\\}\cup\nabla_{xy}\cup N_{x}\cup
N_{y}\cup P_{xy}$

as the _core neighborhood_ of an edge $xy\in E$. It is immediate that the core
neighborhood represents a partition of all vertices with distance less than or
equal to $2$ from $x$ and $y$. Moreover, the Reduction Lemma [3, Lemma 2.3]
shows that when computing the condensed Ricci curvature of an edge $xy\in E$,
it is sufficient to consider only the induced subgraph of $G$ lying within the
core neighborhood $\mathcal{N}_{xy}$.

##  3\. A Rigidity Theorem for Complete Graphs

In this section we establish a rigidity theorem for complete graphs that shows
that a complete graph is the only graph with condensed Ricci curvature
strictly greater than one. Given a connected finite simple graph $G=(V,E)$
with $n$ vertices $v_{1},\dots,v_{n}$, the adjacency matrix $A$ of $G$ is a
symmetric $n\times n$ matrix with entries

$\begin{cases}a_{ij}=1&\text{if}\ v_{i}v_{j}\in E\\\
a_{ij}=0&\text{otherwise}.\end{cases}$

Let $d_{i}$ denote the degree of vertex $v_{i}$ and let
$D=\text{diag}(d_{1},\dots,d_{n})$ be the $n\times n$ diagonal matrix of
degrees. The spectral graph Laplacian is then defined

$L=I-D^{-\frac{1}{2}}AD^{-\frac{1}{2}}.$

With these conventions it follows that the eigenvalues of the spectral graph
Laplacian are nondecreasing with

$0=\lambda_{0}\leq\lambda_{1}\leq\cdots\leq\lambda_{n-1}$

where the multiplicity of the $0$ eigenvalue is the number of connected
components of $G$. Hence, a finite graph is connected if and only if
$\lambda_{1}>\lambda_{0}=0$. Similar to the spectrum of the metric Laplacian
on Riemannian manifolds, the spectrum of the spectral Laplacian on graphs
encodes geometric and structural properties of a graph.

In the context of Riemannian geometry the Lichnerowicz theorem [10] is a
widely celebrated theorem that provides a lower bound for the first nonzero
eigenvalue of the metric Laplacian when the Ricci curvature has a strict
positive lower bound. More precisely, if $(M^{n},g)$ is an $n$-dimensional
compact Riemannian manifold with

$Ric_{g}\geq(n-1)kg$

for some positive constant $k>0$, then the first positive eigenvalue of the
metric Laplacian $\lambda_{1}\geq nk$. The graph analog of the Lichnerowicz
theorem similarly shows that a positive lower bound on condensed Ricci
curvature serves as a lower bound on the first nonzero eigenvalue of the
spectral graph Laplacian.

######  Theorem 3.1.

[11, Theorem 4.2] Suppose $G=(V,E)$ is a connected finite simple graph with
condensed Ricci curvature

$\Bbbk(x,y)\geq\kappa_{0}>0$

for some positive constant $\kappa_{0}$ and all edges $xy\in E$. Then the
first nonzero eigenvalue of the spectral graph Laplacian
$\lambda_{1}\geq\kappa_{0}$.

In order to establish a rigidity theorem for complete graphs we first record
some well known results on the first nonzero eigenvalue of the spectral graph
Laplacian.

######  Theorem 3.2.

[6] Let $G=(V,E)$ be a simple graph on $n$ vertices. Then

$\lambda_{1}\leq\frac{n}{n-1}$

with equality if and only if $G$ is complete. Moreover, complete graphs are
the only graphs with $\lambda_{1}>1$.

We now prove the following rigidity theorem through direct computation and the
aid of Theorems 3.1 and 3.2.

######  Theorem 3.3.

A connected finite simple graph $G=(V,E)$ is complete if and only if the
condensed Ricci curvature $\Bbbk(x,y)>1$ for all edges $xy\in E$.

###### Proof.

Let $G=(V,E)$ be the complete graph on $n$ vertices and let $xy\in E$. Then
the triangle set $\nabla_{xy}=V\setminus\\{x,y\\}$ since $G$ is complete. Let
$m_{x}$ and $m_{y}$ denote $\varepsilon=\frac{1}{2}$ probability measures
based at $x$ and $y$, respectively, defined as in (2.1). Consider the
transport plan $\xi:V\times V\to[0,1]$ between $m_{x}$ and $m_{y}$ given by

$\xi(v,w)=\begin{cases}\frac{1}{2}-\frac{1}{2(n-1)}&v=x,w=y\\\
\frac{1}{2(n-1)}&v=w\\\ 0&\text{otherwise},\end{cases}$

which transports excess mass directly from $x$ to $y$ and leaves the remainder
of the distribution fixed. Then by Definition 2.1 the transport distance

$W(m_{x},m_{y})\leq\sum_{v\in V}\sum_{w\in
V}\rho(v,w)\xi(v,w)=\xi(x,y)=\frac{1}{2}-\frac{1}{2(n-1)}$

and therefore the condensed Ricci curvature

(3.1) $\Bbbk(x,y)=2\kappa_{\frac{1}{2}}(x,y)\geq
2\left(1-\left(\frac{1}{2}-\frac{1}{2(n-1)}\right)\right)=1+\frac{1}{n-1}>1.$

Conversely, let $G=(V,E)$ be a connected finite simple graph with condensed
Ricci curvature $\Bbbk(x,y)>1$ for all edges $xy\in E$. Then by the graph
Lichnerowicz Theorem 3.1 of [11], it follows $\lambda_{1}>1$ and therefore $G$
is complete by Theorem 3.2. ∎

######  Remark 3.4.

Using the $1$-Lipschitz function $f:V\to{\mathbb{R}}$ defined by

$f(v)=\begin{cases}1&v=x\\\ 0&\text{otherwise},\end{cases}$

it follows from the Kantorovich Duality Theorem 2.2 that the transport
distance

$W(m_{x},m_{y})\geq\sum_{v\in
V}f(v)(m_{x}(v)-m_{y}(v))=m_{x}(x)-m_{y}(x)=\frac{1}{2}-\frac{1}{2(n-1)}$

and therefore the condensed Ricci curvature

(3.2) $\Bbbk(x,y)=2\kappa_{\frac{1}{2}}(x,y)\leq
2\left(1-\left(\frac{1}{2}-\frac{1}{2(n-1)}\right)\right)=1+\frac{1}{n-1}.$

Hence, from (3.1) and (3.2) we see that the condensed Ricci of the complete
graph on $n$ vertices is

$\Bbbk(x,y)=1+\frac{1}{n-1}=\frac{n}{n-1}$

for all edges $xy\in E$ as stated in [11, Ex. 1].

##  4\. Strongly Regular Graphs

In this section we derive an explicit formula for the condensed Ricci
curvature of strongly regular graphs $G=(V,E)$ in terms of the graph
parameters and the size of a maximum matching in the core neighborhood. Recall
that a simple, undirected, finite graph $G=(V,E)$ is said to be a strongly
regular graph with parameters $(n,d,\alpha,\beta)$ if $G$ has $n$ vertices of
constant degree $d$ where any two adjacent vertices share $\alpha\geq 0$
common neighbors and any two nonadjacent vertices share $\beta\geq 1$ common
neighbors. Given an edge $xy\in E$, we consider the core neighborhood
decomposition (2.2) given by

$\mathcal{N}_{xy}=\\{x\\}\cup\\{y\\}\cup\nabla_{xy}\cup N_{x}\cup N_{y}\cup
P_{xy}$

that was introduced in [3] and where the neighbor sets

$N_{x}=\Gamma(x)\setminus(\nabla_{xy}\cup\\{y\\})\quad\text{and}\quad
N_{y}=\Gamma(x)\setminus(\nabla_{xy}\cup\\{y\\}).$

Since strongly regular graphs have diameter $2$, it follows that
$V=\mathcal{N}_{xy}$. Hence, for strongly regular graphs the induced subgraph
lying within the core neighborhood of any edge accounts for the entire graph
$G$. Moreover, from strong regularity we see $|\nabla_{xy}|=\alpha$ so that

$|N_{x}|=|N_{y}|=d-\alpha-1\quad\text{and}\quad|P_{xy}|=n-2d-\alpha.$

Our strategy in the derivation of an explicit formula for the condensed Ricci
curvature of strongly regular graphs reformulates the problem of finding
candidates for an optimal transport plan and an optimal Kantorovich potential
into a maximum matching problem.

######  Definition 4.1.

A matching on a finite undirected simple graph $G=(V,E)$ is a subset of edges
$\mathcal{M}\subseteq E$ such that no vertex in $V$ is incident to more than
one edge in $\mathcal{M}$. The size of a matching $\mathcal{M}$ on $G$ is the
total number of edges in $\mathcal{M}$. A matching $\mathcal{M}$ on $G$ is
said to be a maximum matching if it contains the largest possible number of
edges. A maximum matching between disjoint subsets of vertices $S,T\subset V$
is said to be a perfect matching if every vertex in $S$ and $T$ is incident to
exactly one edge of the matching.

Let $G=(V,E)$ be strongly regular graph with edge $xy\in E$. Given a maximum
matching $\mathcal{M}$ between the neighbor sets $N_{x}$ and $N_{y}$, we
denote the vertices that are matched in $N_{x}$ and $N_{y}$ by $M_{x}$ and
$M_{y}$, respectively. If $\mathcal{M}$ is not a perfect matching we denote
the remaining unmatched vertices in $N_{x}$ and $N_{y}$ by
$U_{x}=N_{x}\setminus M_{x}$ and $U_{y}=N_{y}\setminus M_{y}$, respectively.
With this decomposition of $N_{x}$ and $N_{y}$ into matched and unmatched
vertices it is fairly straightforward to develop a candidate for an optimal
transport plan. However, developing a candidate for an optimal Kantorovich
potential requires a deeper understanding of the incidence relations between
edges in a given maximum matching and the remaining edges between $N_{x}$ and
$N_{y}$.

######  Definition 4.2.

Let $\mathcal{M}$ be a matching on an undirected simple graph $G=(V,E)$. An
alternating path $P$ in $G$ is a path $P=v_{0}v_{1}\cdots v_{k}$ such that the
initial vertex $v_{0}$ is unmatched and $v_{i}v_{i+1}\in\mathcal{M}$ if and
only if $v_{i-1}v_{i}\not\in\mathcal{M}$ for all $0<i<k$. An augmenting path
$P$ in $G$ is an alternating path $P=v_{0}v_{1}\cdots v_{k}$ such that the
terminal vertex $v_{k}$ is also unmatched.

We refer to a single vertex as a trivial path. Hence, with this convention any
unmatched vertex is by itself a trivial augmenting path. One of the main tools
in our construction of a candidate for an optimal Kantorovich potential is
Berge’s Lemma [2].

######  Lemma 4.3 (Berge’s Lemma).

A matching $\mathcal{M}$ on an undirected simple graph $G=(V,E)$ is a maximum
matching if and only if $G$ contains no nontrivial augmenting paths.

Given a matching $\mathcal{M}$ on a graph $G=(V,E)$ and a subset of vertices
$U\subseteq V$, we denote the subset of matched vertices in $U$ by
$\mathcal{M}(U)$. For a subset of vertices $S\subseteq V$ we denote the subset
of alternating paths with initial vertex in $S$ by $A_{S}$. For subsets of
vertices $S,T\subseteq V$ we denote the set of vertices in $T$ that lie along
an alternating path initiated in $S$ by $A_{S}(T)$. We use the contrapositive
of the following lemma to show that certain vertices in the neighbor sets
$N_{x}$ and $N_{y}$ have no edges between them. Ultimately, this allows us to
ensure our candidate for an optimal Kantorovich potential is $1$-Lipschitz.

######  Lemma 4.4.

Let $H=(V,E)$ be a finite, undirected, bipartite graph with parts $S,T$. Let
$\mathcal{M}$ be a maximum matching on $H$ and suppose $v\in A_{S}(S)$ and
$w\in T$. If $vw\in E$, then $w\in A_{S}(T)$.

###### Proof.

First note that since $H$ is bipartite, it follows that an alternating path
initiated in $S$ can only terminate in an unmatched vertex in $T$. Therefore,
if $v\in A_{S}(S)$ is an unmatched vertex, then $v$ must be the initial vertex
of an alternating path. But then $w$ must be a matched vertex so that $w\in
A_{S}(T)$ since otherwise $vw\in E$ is an augmenting path, contradicting
Berge’s Lemma as $\mathcal{M}$ is a maximum matching. Now assume that $v$ is a
matched vertex and that $P=v_{0}v_{1}\dots v_{i}v$ is an alternating path
initiated at $v_{0}\neq v\in S$ that terminates at $v\in S$. Then since
$v_{0}\in S$ is an unmatched vertex and $H$ is bipartite, it follows that
$v_{i}v\in\mathcal{M}$ since $P$ is an alternating path. But then
$vw\not\in\mathcal{M}$ since otherwise $v$ is incident to more than one edge
in $\mathcal{M}$. But then the concatenation $P^{\prime}=v_{0}v_{1}\dots
v_{i}vw$ is an alternating path so that $w\in A_{S}(T)$. ∎

We use this final lemma to compute a lower bound for condensed Ricci curvature
from our Kantorovich potential in terms of the number of vertices in $N_{x}$
and $N_{y}$ along alternating paths initiated in $N_{y}$.

######  Lemma 4.5.

Let $H=(V,E)$ be a finite, undirected, bipartite graph with parts $S,T$.
Suppose $\mathcal{M}$ is a maximum matching of size $m$ on $H$. Then

$|A_{S}(S)|=|A_{S}(T)|+|S|-m.$

###### Proof.

Let $v\in A_{S}(T)$ and suppose $P=v_{0}v_{1}\cdots v_{i}v$ is an alternating
path initiated at $v_{0}\in S$ that terminates at $v\in T$. Then since
$v_{0}\in S$ is an unmatched vertex and $H$ is bipartite, it follows that
$v_{i}v\not\in\mathcal{M}$ since $P$ is an alternating path. If $v$ is an
unmatched vertex, then $P$ is an augmenting path contrary to Berge’s Lemma as
$\mathcal{M}$ is a maximum matching. Hence, $v$ must be a matched vertex and
therefore since matched vertices are incident to a single edge in the
matching, it follows that there exits a unique vertex $w\in S$ such that
$vw\in\mathcal{M}$. But then the concatenation $P^{\prime}=v_{0}v_{1}\cdots
vw$ is an alternating path initiated in $S$ that terminates at the matched
vertex $w\in\mathcal{M}(S)$. Hence, for each $v\in A_{S}(T)$ there is a unique
$w\in A_{S}(S)\cap\mathcal{M}(S)$ with $vw\in\mathcal{M}$.

On the other hand, given $w\in A_{S}(S)\cap\mathcal{M}(S)$ suppose
$P=w_{0}w_{1}\cdots w_{i}w$ is an alternating path initiated in $S$ that
terminates at $w\in S$. Then since $w$ is a matched vertex by assumption, it
is clear that $P$ is a nontrivial path and that $w_{i}\in T$ is the unique
matched vertex with $w_{i}w\in\mathcal{M}$. But then
$P^{\prime}=w_{0}w_{1}\cdots w_{i}$ is an alternating path initiated in $S$
that terminates at $w_{i}\in T$ so that $w_{i}\in A_{S}(T)$. Hence, for each
$w\in A_{S}(S)\cap\mathcal{M}(S)$ there is a unique $v\in A_{S}(T)$ with
$vw\in\mathcal{M}$.

Thus, since $A_{S}(T)\subseteq\mathcal{M}(T)$, we have a bijection between the
subsets $A_{S}(T)$ and $A_{S}(S)\cap\mathcal{M}(S)$ and therefore
$|A_{S}(T)|=|A_{S}(S)\cap\mathcal{M}(S)|$. Noting
$S\setminus\mathcal{M}(S)\subseteq A_{S}(S)$ since under our conventions all
unmatched vertices in $S$ are trivial alternating paths initiated in $S$, it
follows

$\displaystyle|A_{S}(T)|$
$\displaystyle=|A_{S}(S)\cap\mathcal{M}(S)|=|A_{S}(S)\setminus(S\setminus\mathcal{M}(S))|$
$\displaystyle=|A_{S}(S)|-|S\setminus\mathcal{M}(S)|=|A_{S}(S)|-(|S|-m).$

Hence,

$|A_{S}(S)|=|A_{S}(T)|+|S|-m$

as desired. ∎

We are now in a position to derive an explicit formula for condensed Ricci
curvature of an edge $xy\in E$ in terms of the graph parameters and the size
of a maximum matching between the neighbor sets $N_{x}$ and $N_{y}$.

######  Theorem 4.6.

Let $G=(V,E)$ be strongly regular graph with parameters $(n,d,\alpha,\beta)$.
Suppose $xy\in E$ with maximum matching $\mathcal{M}$ of size $m$ between
$N_{x}$ and $N_{y}$. Then the condensed Ricci curvature

$\Bbbk(x,y)=\frac{\alpha+2}{d}-\frac{|N_{x}|-m}{d}.$

###### Proof.

Let $xy\in E$ and consider the core neighborhood decomposition of $G$ as in
(2.2). Let $H$ denote the induced bipartite subgraph consisting of all edges
in $E$ between vertices in $N_{x}$ and $N_{y}$. Suppose that $\mathcal{M}$ is
a maximum matching on $H$ of size $|\mathcal{M}|=m$. Assuming $\mathcal{M}$ is
not a perfect matching, as above we denote the matched and unmatched vertices
in $N_{x}$ and $N_{y}$ by $M_{x},M_{y}$ and $U_{x},U_{y}$, respectively. Note
that $|U_{x}|=|U_{y}|$ since $|N_{x}|=|N_{y}|$ and $|M_{x}|=|M_{y}|$. For
$i=1,\dots,|N_{x}|-m$, let $x_{i}^{u}\in N_{x}$ and $y_{i}^{u}\in N_{y}$
denote the unmatched vertices in $H$ and for $j=1,\dots,m$, let $x_{j}^{m}\in
N_{x}$ and $y_{j}^{m}\in N_{y}$ denote the matched vertices in $H$. Up to
reordering of indices we may assume $x_{i}^{m}y_{i}^{m}\in\mathcal{M}$ for
$i=1,\dots,m$.

Clearly, there are no edges in $E$ incident to unmatched vertices since
$\mathcal{M}$ is a maximum matching so $x_{i}^{u}y_{j}^{u}\not\in E$ for all
$1\leq i,j\leq|N_{x}|-m$. But then $\rho(x_{i}^{u},y_{i}^{u})=2$ in $G$ since
$x_{i}^{u}$ and $y_{i}^{u}$ are not adjacent and $G$ has diameter $2$. Hence,
for $1\leq i\leq|N_{x}|-m$, we may pair unmatched vertices $x_{i}^{u}$ with
$y_{i}^{u}$ along $2$-step paths in $G$. For simplicity in notation write
$x_{i}^{u}y_{i}^{u}\in\mathcal{M}^{2}$ to denote unmatched vertices
$x_{i}^{u}\in U_{x}$ and $y_{i}^{u}\in U_{y}$ that have been paired along a
$2$-step path. Together these matchings/pairings induce a transport plan that
moves excess mass from $x$ to $y$ and that moves mass from $x_{i}^{m}\in
M_{x}$, $x_{i}^{u}\in U_{x}$ directly to their matched/paired neighbors
$y_{i}^{m}\in M_{y}$, $y_{i}^{u}\in U_{y}$ along $1$-step and $2$-step paths,
respectively. More precisely, consider the transport plan defined by

(4.1) $\xi(v,w)=\begin{cases}\frac{1}{2}-\frac{1}{2d}&v=x,w=y\\\
\frac{1}{2d}&v=w\in\nabla_{xy}\cup\\{x\\}\cup\\{y\\}\\\
\frac{1}{2d}&vw\in\mathcal{M}\\\ \frac{1}{2d}&vw\in\mathcal{M}^{2}\\\
0&\text{otherwise}.\end{cases}$

Noting that $|N_{x}|=|N_{y}|=d-\alpha-1$ it follows that the transport
distance

$\displaystyle W(m_{x},m_{y})\leq\sum_{v\in V}\sum_{w\in V}\rho(v,w)\xi(v,w)$
$\displaystyle=\left(\frac{1}{2}-\frac{1}{2d}\right)+m\cdot\frac{1}{2d}+\left(|N_{x}|-m\right)\cdot
2\cdot\frac{1}{2d}$
$\displaystyle=\frac{1}{2}+\frac{1}{2d}\left((d-\alpha-2)+(|N_{x}|-m)\right).$

Thus, the condensed Ricci curvature

(4.2) $\displaystyle\Bbbk(x,y)=2\kappa_{\frac{1}{2}}(x,y)$ $\displaystyle\geq
2\left(1-\left(\frac{1}{2}+\frac{1}{2d}\left((d-\alpha-2)+(|N_{x}|-m)\right)\right)\right)$
$\displaystyle=\frac{\alpha+2}{d}-\frac{|N_{x}|-m}{d}.$

Next we define a Kantorovich potential function $f:V\to{\mathbb{R}}$ with
respect to the matching $\mathcal{M}$ by

(4.3) $f(z)=\begin{cases}1&z=x\\\ 1&z\in N_{x}\setminus A_{N_{y}}(N_{x})\\\
-1&z\in A_{N_{y}}(N_{y})\\\ 0&\text{otherwise}\end{cases}$

where $A_{N_{y}}(N_{x})$ and $A_{N_{y}}(N_{y})$ denote the vertices in $N_{x}$
and $N_{y}$, respectively, that lie along alternating paths initiated in
$N_{y}$. Taking $T=N_{x}$ and $S=N_{y}$ in Lemma 4.4, it follows that there
are no edges between $N_{x}\setminus A_{N_{y}}(N_{x})$ and $A_{N_{y}}(N_{y})$.
Hence, $f$ is a $1$-Lipschitz function on $V$.

Let $a_{x}=|A_{N_{y}}(N_{x})|$ and $a_{y}=|A_{N_{y}}(N_{y})|$. Then

$|N_{x}\setminus A_{N_{y}}(N_{x})|=|N_{x}|-a_{x}=d-\alpha-1-a_{x}$

and by Lemma 4.5

$a_{y}=|A_{N_{y}}(N_{y})|=|A_{N_{y}}(N_{x})|+|N_{y}|-m=a_{x}+|N_{x}|-m.$

Therefore, by the Kantorovich duality theorem, we find that the transport
distance

$\displaystyle W(m_{x},m_{y})$ $\displaystyle\geq\sum_{z\in
V}f(z)(m_{x}(z)-m_{y}(z))=\left(\frac{1}{2}-\frac{1}{2d}\right)+(d-\alpha-1-a_{x})\cdot\frac{1}{2d}+a_{y}\cdot\frac{1}{2d}$
$\displaystyle=\left(\frac{1}{2}-\frac{1}{2d}\right)+(d-\alpha-1-a_{x})\frac{1}{2d}+(a_{x}+|N_{x}|-m)\frac{1}{2d}$
$\displaystyle=\frac{1}{2}+\frac{1}{2d}\left((d-\alpha-2)+(|N_{x}|-m)\right).$

Similar to the calculation (4.2), we have

(4.4)
$\Bbbk(x,y)=2\kappa_{\frac{1}{2}}(x,y)\leq\frac{\alpha+2}{d}-\frac{|N_{x}|-m}{d}.$

Hence, from the inequalities (4.2) and (4.4), it follows

$\Bbbk(x,y)=\frac{\alpha+2}{d}-\frac{|N_{x}|-m}{d}.$

Now if $\mathcal{M}$ is a perfect matching between $N_{x}$ and $N_{y}$, then
the size of $\mathcal{M}$ is by definition $m=|N_{x}|=|N_{y}|=d-\alpha-1$.
Moreover, since all vertices in $N_{x}$ and $N_{y}$ are matched by some edge
in $\mathcal{M}$, it follows that there are no alternating paths initiated in
$N_{x}$ or $N_{y}$ in the induced bipartite subgraph between vertices in
$N_{x}$ and $N_{y}$. But then $A_{N_{y}}(N_{x})=\varnothing$ and
$A_{N_{y}}(N_{y})=\varnothing$. Taking all of this into account, we see that
the same transport plan (4.1) coupled with the Kantorovich potential (4.3)
yield the equality

$\Bbbk(x,y)=\frac{\alpha+2}{d}$

in the case of a perfect matching between $N_{x}$ and $N_{y}$. ∎

###  4.1. Strongly Regular Graphs of Girths 4 and 5

In this subsection we exploit the adjacency properties of strongly regular
graphs $G=(V,E)$ to derive explicit formulas for the condensed Ricci curvature
of strongly regular graphs of girths $4$ and $5$. Let $G=(V,E)$ be a strongly
regular graph with parameters $(n,d,\alpha,\beta)$. It is well known [5] that
if $G$ has girth $5$, then $\alpha=0$ and $\beta=1$, while a strongly regular
graph $G$ with girth $4$ has $\alpha=0$ and $\beta\geq 2$. The following
theorem is an immediate consequence of Theorem 4.6 and the fact that the girth
of a graph is its minimal cycle length.

######  Theorem 4.7.

Let $G=(V,E)$ be strongly regular graph with parameters $(n,d,\alpha,\beta)$.
If the girth of $G$ is $5$, then the condensed Ricci curvature

$\Bbbk(x,y)=\frac{3}{d}-1$

for all edges $xy\in E$.

###### Proof.

Let $xy\in E$ and consider the core neighborhood decomposition of $G$ as in
(2.2). Let $x_{0}\in N_{x}$ and suppose that $x_{0}y_{0}\in E$ for some
$y_{0}\in N_{y}$. Then $xx_{0}y_{0}yx$ is a cycle of length $4$, contradicting
the fact that $G$ has girth $5$. Thus, there are no edges between $N_{x}$ and
$N_{y}$ so the maximum matching between $N_{x}$ and $N_{y}$ has size $m=0$.
Therefore, from Theorem 4.6 with $\alpha=0$ and $m=0$, we find that the
condensed Ricci curvature

$\Bbbk(x,y)=\frac{3}{d}-1.\vspace{-0.85cm}$

∎

######  Remark 4.8.

The transportation plan (4.1) and Kantorovich potential (4.3) can be used to
compute this curvature formula directly. Since $G$ is girth $5$, there are no
edges between vertices in $N_{x}$ and $N_{y}$ so $\mathcal{M}=\varnothing$ and
each vertex in $N_{x}$ can be paired to a vertex in $N_{y}$ through a $2$-step
path. Moreover, if we consider induced bipartite subgraph $H$ between $N_{x}$
and $N_{y}$, the collection of alternating paths in $H$ that start in $N_{y}$
consists solely of trivial paths so $A_{N_{y}}(N_{y})=N_{y}$ and
$A_{N_{y}}(N_{x})=\varnothing$.

In the girth 4 case we appeal to Hall’s theorem, which can be used to give
necessary and sufficient conditions for perfect matchings on bipartite graphs.
Let $H=(V,E)$ be a finite, bipartite graph with parts $S,T$. For any subset
$W\subseteq S$, we define

$\Gamma_{T}(W)=\\{v\in T\ \big{|}\ vw\in E\ \text{for some}\ w\in W\\}$

to be the collection of all vertices $v\in T$ sharing an edge with some vertex
in $W$. More simply put, $\Gamma_{T}(W)$ is the neighbor set of $W$ in $T$.

######  Theorem 4.9 (Hall’s Theorem).

Let $H=(V,E)$ be a finite, undirected, bipartite graph with parts $S,T$. Then
there is a matching that covers $S$ if and only if

$|{\Gamma_{T}(W)}|\geq|{W}|$

for every subset of vertices $W\subseteq S$.

With Hall’s theorem in hand we are able to establish a perfect matching
between the neighbor sets $N_{x}$ and $N_{y}$ for strongly regular graphs with
girth 4 as a consequence of the pigeonhole principle.

######  Theorem 4.10.

Let $G=(V,E)$ be strongly regular graph with parameters $(n,d,\alpha,\beta)$.
If the girth of $G$ is $4$, then the condensed Ricci curvature

$\Bbbk(x,y)=\frac{2}{d}$

for all edges $xy\in E$.

###### Proof.

Let $xy\in E$ and consider the core neighborhood decomposition of $G$ as in
(2.2). Then since $\alpha=0$, it follows that $G$ is triangle free. Suppose
that $x_{i}\in N_{x}$. Then since $x_{i}$ is nonadjacent to $y$, it follows
$x_{i}$ and $y$ have $\beta\geq 2$ common neighbors. Clearly, one such
neighbor is $x$ so $x_{i}$ has $\beta-1\geq 1$ neighbors in $N_{y}$ since
$\nabla_{xy}=\varnothing$. Similarly, each vertex $y_{j}\in N_{y}$ has
$\beta-1\geq 1$ neighbors in $N_{x}$.

Now consider the induced bipartite subgraph $H=G[N_{x}\cup N_{y}]$ consisting
of all edges in $E$ between vertices in $N_{x}$ and $N_{y}$. For sake of
contradiction suppose that Let $X\subseteq N_{x}$ with
$|X|=q>|\Gamma_{N_{y}}(X)|=k$. Since each vertex $x_{i}\in X\subseteq N_{x}$
has $\beta-1\geq 1$ neighbors in $N_{y}$, it follows that there are
$(\beta-1)q$ edges between $X$ and $\Gamma_{N_{y}}(X)$. Therefore, by the
generalized pigeonhole principle there exists some $w\in\Gamma_{N_{y}}(X)$
with

$|\Gamma(w)|\geq\left\lceil\frac{(\beta-1)q}{k}\right\rceil>\left\lceil\frac{(\beta-1)q}{q}\right\rceil=\beta-1$

since $q>k$. This contradicts the fact that every vertex in $N_{y}$ has
exactly $(\beta-1)$ neighbors in $N_{x}$. Thus, $|X|\leq|\Gamma_{N_{y}}(X)|$
for all subsets $X\subset N_{x}$ and therefore since $|N_{x}|=|N_{y}|$ it
follows that there is a perfect matching in $H$ by Hall’s Theorem. Noting that
$\alpha=0$, it follows from Theorem 4.6 with $\alpha=0$ and $m=|N_{x}|$ that
the condensed Ricci curvature

$\Bbbk(x,y)=\frac{2}{d}.\vspace{-0.85cm}$

∎

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