# A Didactic Journey from Statistical Physics to Thermodynamics

Michael Riedl<EMAIL_ADDRESS>Institute for Theoretical Physic -
Johannes Kepler Universität, Altenbergerstr. 69, A-4040 Linz, Austria  Mario
Graml<EMAIL_ADDRESS>School of Education - Johannes Kepler Universität,
Altenbergerstr. 69, A-4040 Linz, Austria

###### Abstract

This paper offers a pedestrian guide from the fundamental properties of
entropy to the axioms of thermodynamics, which are a consequence of the axiom
of statistical physics. It also dismantles flawed concepts, such as assigning
physical meaning to Lagrange multipliers and numerous others. This work also
provides a comprehensive understanding of the Legendre transform via
geometrical, mathematical and physical insights, as well as its connection to
the experimental setup. The central result of this paper is the comprehensive
formalisation of key concepts, including ensembles, variable dependencies,
potentials and natural variables. Furthermore, the framework of
thermodynamics, the state function and the Euler inequality are rigorously
proven from the axiom of statistical physics.

††preprint: APS/123-QED

##  I Introduction

In the framework of statistical physics, physical phenomena are only
expressible through probabilities [1, 2]. Central to this framework is the
concept of entropy [3, 4, 5, 6, 7], a fundamental quantity that encapsulates
the average amount of missing information or surprise associated with a
system’s state. Through the lens of quantum mechanics, these probabilities are
encoded in what is known as a mixed state, which is a mixture of pure states,
each weighted by their respective probabilities [8, 9, 10]. This work aims to
give a pedestrian guide as mathematically precise as needed to develop the
framework starting from entropy, and ending in phenomenological
thermodynamics. It allows us to define precisely the heavily misused terms in
the context.
In sec. II we present the basic ideas of entropy, leading to the definition of
entropy, and then demonstrate all its properties in its special tailor-made
example. Entropy is determined in sec. III with the mathematical formulation
of the physical intuition on the given experimental data. A comprehensive
discussion of the connection between theory and experiment is given in sec. IV
and is concluded with the stability conditions and Maxwell relations. Changing
the tunable parameters in the experimental setup must be incorporated into the
theory, as discussed in section sec. V. This will be done threw a full
exploration of the Legendre transformation in terms of its geometrical
meaning, mathematical formulation and physical consequences for the
experimental setup. The principal findings, as detailed in sec. VI, are the
precise formulation and definition of ensembles, their representation,
potentials and natural variables. Finally, in sec. VII, the framework of
phenomenological thermodynamics, including the state function and Euler
inequality, is derived from the axiom of statistical physics.

##  II Idea and Definition of Entropy

In statistical physics, physical concepts are described using statistics and
probabilities. In the framework of quantum mechanics, these probabilities are
encoded in the eigenvalues of the mixed state $\hat{\mathrm{\rho}}$ (hereafter
referred to as the state). The state consists of pure states
$\hat{\mathrm{\rho}}_{i}$ with associated probabilities $p_{i}$. The objective
is to approximate the state by mathematically determining the $p_{i}$ as
accurately as possible. The approximate state is quantified by a number that
satisfies the following three conditions: a value of zero represents complete
knowledge of the state, while the maximum value represents complete
uncertainty. When quantifying independent systems together, the numbers of the
source systems should add up. The operator $\ln(\hat{\mathrm{\rho}})$ fulfils
these three properties of quantifying the state of a system. The resulting
functional $S[\hat{\mathrm{\rho}}]$ is known as von Neumann-entropy.

Definition II.1:  Entropy The entropy $S[\hat{\mathrm{\rho}}]$ for a given
state $\hat{\mathrm{\rho}}$ is defined as $\displaystyle
S[\hat{\mathrm{\rho}}]\coloneqq-k\Tr\left(\hat{\mathrm{\rho}}\,\ln(\hat{\mathrm{\rho}})\right)$
(1) where $k>0$ is a proportionality constant.

Entropy is a measure of the average missing information. It can also be
interpreted as a measure of how surprised someone would be at the result of an
experiment. The following examples will clarify the intuitions regarding
surprise and average missing information.

Complete knowledge: If the state is fully known, the inner product
$\innerproduct{\phi_{i}}{\hat{\mathrm{\rho}}_{j}(\phi_{i})}=\delta_{ij}$ leads
to

$S[\hat{\mathrm{\rho}}]=-k\Tr\left(\hat{\mathrm{\rho}}\ln\left(\hat{\mathrm{\rho}}\right)\right)=-k\sum\limits_{i}\delta_{ij}\ln\left(\delta_{ij}\right)=0$
(2)

Eq. (2) illustrates that there is no surprise, if the experiment always
determines the pure state $\hat{\mathrm{\rho}}_{j}$.

No knowledge: On the other hand, to obtain a maximally surprising result from
the experiment, no prior knowledge about the state is assumed. This implies
that the state is consisting of uniformly distributed pure states. Assuming
that the state consists of $n$ such pure states, then

$S[\hat{\mathrm{\rho}}]=-k\Tr\left(\hat{\mathrm{\rho}}\ln\left(\hat{\mathrm{\rho}}\right)\right)=-k\sum\limits_{j=1}^{n}\frac{1}{n}\ln\left(\frac{1}{n}\right)=k\ln(n)$
(3)

Eq. (3) represents the maximum value of $S$ and is known as the Boltzmann
entropy, with $n$ being the number of uniformly distributed pure states. The
proof of the third property is presented in the appendix A, where it is shown
that for two independent systems A and B, the entropy is additive:

$S[\hat{\mathrm{\rho}}_{\text{AB}}]=S[\hat{\mathrm{\rho}}_{\text{A}}]+S[\hat{\mathrm{\rho}}_{\text{B}}]$
(4)

To gain an understanding of the three properties and how to handle them, let
us consider an example:

Three coin tosses
In this thought experiment, we will toss a coin three times. There are only
two possible outcomes for each toss: Heads (H) or Tails (T). There are
$n=2^{3}$ different outcomes for tossing a coin three times.

Figure 1: Three coin tosses with the single outcomes Heads (H) or Tails (T) is
represented. The probability for the single outcome H is given by
$p_{\text{H}}$ and the probability for the single outcome T is given by
$p_{\text{T}}$ respective. The three tosses resulting in $2^{3}$ different
possibilities for the pure state, denoted as HHH, HHT, …, TTT.

This experimental setup is illustrated in fig. 1. In this setup, we choose
$k=\frac{1}{\ln 2}$, so the entropy will be in units of $\log_{2}$, also known
as bits of information. We will consider three different scenarios:

Scenario one involves three fair coin tosses:
Assuming no prior knowledge of the coin, we can infer that the coin is fair,
with an equally distribution of H and T. In order to ascertain the state, it
is necessary to answer three questions of either H or T. As all pure states
are equally distributed, eq. (3) of the Boltzmann entropy can be used to
calculate the average missing information, denoted as the entropy:

$S[\hat{\mathrm{\rho}}]=\log_{2}(n)=\log_{2}\left(2^{3}\right)=3$ (5)

In eq. (5), the average missing information, expressed in units of bits, is
three, corresponding to the three answers of H or T.

Scenario two consists of three highly skewed coin tosses:
In this instance, we assume that the coin is biased, with a probability of H,
$p_{\text{H}}=0.95$. Therefore, we now have more information about the system
than we did previously. Since the pure states are not equally distributed, we
can no longer use Boltzmann’s entropy. However, the state still consists of
three independent subsystems, the three coin flips, so we can still use the
third property eq. (4) and add up the subsystems:

$\displaystyle S[\hat{\mathrm{\rho}}_{AAA}]$
$\displaystyle=S[\hat{\mathrm{\rho}}_{A}]+S[\hat{\mathrm{\rho}}_{A}]+S[\hat{\mathrm{\rho}}_{A}]=3S[\hat{\mathrm{\rho}}_{A}]$
$\displaystyle=-3\left(p_{\text{H}}\log_{2}\left(p_{\text{H}}\right)+p_{\text{T}}\log_{2}\left(p_{\text{T}}\right)\right)\approx
0.86$ (6)

The average missing information is 0.86 bits, and one is not so surprised
because one can expect two to three H with a probability of $>99$%.

Scenario three uses a memory coin:
The difference between this setup and the previous two is that the tosses are
dependent on each other. The additional rule for this experiment is that the
probability of getting the same result as the previous flip is halved.
Starting with a fair toss, the probabilities of the pure states are calculated
as follows

  * •

$p(H,H,H)=p(T,T,T)=\frac{1}{2}\cdot\frac{1}{4}\cdot\frac{1}{8}=\frac{1}{64}$

  * •

$p(H,H,T)=p(T,T,H)=\frac{1}{2}\cdot\frac{1}{4}\cdot\frac{7}{8}=\frac{7}{64}$

  * •

$p(T,H,H)=p(H,T,T)=\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{3}{8}=\frac{9}{64}$

  * •

$p(T,H,T)=p(H,T,H)=\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{8}=\frac{15}{64}$

To calculate the entropy for dependent systems we have to use the general
formula eq. (1):

$\displaystyle
S[\hat{\mathrm{\rho}}]=-\Tr\left(\hat{\mathrm{\rho}}\log_{2}\left(\hat{\mathrm{\rho}}\right)\right)=-\sum\limits_{i=1}^{8}p_{i}\log_{2}(p_{i})\approx
2.7<3$ (7)

As can be seen in eq. (7), one is still surprised, but due to the information
about the switching probabilities, the surprise is not as great as the initial
value of 3.

##  III Determining the Entropy from Experimental Data

As we have seen, entropy allows us to quantify our knowledge of a state. In
practice, however, we cannot determine the distribution of the state and can
only measure certain quantities of the system. From these experimental
results, we want to extract the probability distribution of the state in the
most unbiased way. This leads to the axiom of statistical physics:

Axiom :  Axiom of Statistical Physics If we cannot, or do not want to, know
the state of a system completely, then we have to construct the probability
distribution of the state, such that the state 1.) reproduces the experimental
measurements 2.) is maximally unbiased.

This axiom can now be equivalently stated:

Axiom :  Equivalent Formulation Let $\hat{\mathrm{\rho}}$ be the unknown state
of the system, we want to find $\hat{\mathrm{\rho}}$ such that 1.)
$\expectationvalue{\hat{\mathrm{R}}_{i}}_{\hat{\mathrm{\rho}}}=r_{i}\Leftrightarrow\Tr\left(\hat{\mathrm{\rho}}\hat{\mathrm{R}}_{i}\right)=r_{i}$
and 2.) $S[\hat{\mathrm{\rho}}]$ is maximal. where $\hat{\mathrm{R}}_{i}$
denotes the $i^{\text{th}}$ observable, $r_{i}\in\mathbb{R}$ is the
experimental measurement of $\hat{\mathrm{R}}_{i}$.

The first part of the axiom ensures that the constructed state reflects the
measured values $r_{i}$. The second part of the axiom ensures maximum
ignorance of the unknown information, resulting in the maximization of the
entropy.
This consideration leads to a variation of $S[\hat{\mathrm{\rho}}]$ with
boundary conditions. We solve this problem using the method of Lagrange
multipliers. The first part of the axiom gives us the number of Lagrange
multipliers $k\lambda^{\prime}$. Additionally, the property of the state
$\Tr(\hat{\mathrm{\rho}})=1$ provides us with an additional Lagrange
multiplier $k\Omega^{\prime}$:

$\displaystyle
S[\hat{\mathrm{\rho}}]=-k\Tr\left(\hat{\mathrm{\rho}}\ln(\hat{\mathrm{\rho}})\right)+$
$\displaystyle k\Omega^{\prime}\left(\Tr(\hat{\mathrm{\rho}})-1\right)$
$\displaystyle-\sum\limits_{j=1}^{J}$ $\displaystyle
k\lambda_{j}^{\prime}\bigg{(}\Tr(\hat{\mathrm{R}}_{j}\hat{\mathrm{\rho}})-r_{j}\bigg{)}$
(8a) For practical reasons, we scale the multipliers with $k$ so that the
variation is independent of $k$. Carrying out the variation of eq. (8a) and
setting it to zero, as required by the method
$\forall\hat{\mathrm{\tau}}:\hskip
5.0pt0\stackrel{{\scriptstyle!}}{{=}}\delta_{\tau}S[\hat{\mathrm{\rho}}]\coloneqq\lim_{\epsilon\rightarrow
0}\frac{S[\hat{\mathrm{\rho}}+\epsilon\hat{\mathrm{\tau}}]-S[\hat{\mathrm{\rho}}]}{\epsilon}$
(8b) leads to the condition
$0=-k\Tr\left(\hat{\mathrm{\tau}}\Bigg{(}\ln\left(\hat{\mathrm{\rho}}\right)-\Omega+\sum_{j=1}^{J}\lambda_{j}^{\prime}\hat{\mathrm{R}}_{j}\Bigg{)}\right)$
(8c) with $\Omega=\Omega^{\prime}-1$. In accordance with eq. (8b), eq. (8c)
must hold for all $\hat{\mathrm{\tau}}$, leading to a more refined condition:
$\ln\left(\hat{\mathrm{\rho}}\right)-\Omega+\sum_{j=1}^{J}\lambda_{j}^{\prime}\hat{\mathrm{R}}_{j}=0$
(8d) From the resultant expression in eq. (8d), we derive an equation for
$\hat{\mathrm{\rho}}$:
$\hat{\mathrm{\rho}}(\Omega,\lambda^{\prime})=\exp\left(\Omega-\sum_{j=1}^{J}\lambda_{j}^{\prime}\hat{\mathrm{R}}_{j}\right)$
(8e) where we introduced the shorthand
$\lambda^{\prime}=\left(\lambda_{1}^{\prime},\lambda_{2}^{\prime},\dots,\lambda_{J}^{\prime}\right)$.
The final step to complete the variation principle is to determine the
Lagrange multipliers $\lambda^{\prime}$ and $\Omega$. For this purpose, we
utilize our boundary conditions. Using the condition
$\Tr(\hat{\mathrm{\rho}})=1$ fixes $\Omega$ to
$\Omega(\lambda^{\prime})=-\ln\left(\Tr\Bigg(\exp\bigg(-\sum_{j=1}^{J}\lambda_{j}^{\prime}\hat{\mathrm{R}}_{j}\bigg{missing})\Bigg{missing})\right)$
(8f) To derive the remaining $J$ equations, we leverage the property of the
method of the Lagrangian parameters:
$\partialderivative{S(\lambda^{\prime},\Omega)}{\lambda^{\prime}_{i}}=0$ (8g)
Calculating the entropy with our derived $\hat{\mathrm{\rho}}$ in eq. (8e)
yields
$S(\lambda^{\prime})=-k\Omega(\lambda^{\prime})+k\sum\limits_{j=1}^{J}\lambda_{j}^{\prime}r_{j}$
(8h) This results in $J$ equations, compactly written as
$\frac{\partial\Omega(\lambda^{\prime})}{\partial\lambda_{i}^{\prime}}=r_{i}$
(8i) This system of equations needs to be solved to find
$\lambda^{\prime}(r)$, where $r=(r_{1},r_{2},\dots,r_{J})$.

Substituting these expressions into eqs. (8e) and (8f), we obtain the
following results:

$\hat{\mathrm{\rho}}\left(\lambda^{\prime}\left(r\right)\right)=\exp\left(\Omega(\lambda^{\prime}(r))-\sum_{j=1}^{J}\lambda^{\prime}_{j}(r)\hat{\mathrm{R}}_{j}\right)$
(9)

and

$\Omega(\lambda^{\prime}(r))=-\ln\left(\Tr\Bigg(\exp\bigg(-\sum_{j=1}^{J}\lambda_{j}^{\prime}(r)\hat{\mathrm{R}}_{j}\bigg{missing})\Bigg{missing})\right)$
(10)

We have constructed our state as given in eq. (9) with eq. (10) to ensure that
the resulting state $\hat{\mathrm{\rho}}$ satisfies the axiom of statistical
physics. The associated entropy is given by

$S(r)\equiv
S(\lambda^{\prime}(r))=-k\Omega\left(\lambda^{\prime}\left(r\right)\right)+k\sum_{j=1}^{J}\lambda^{\prime}_{j}\left(r\right)r_{j}$
(11)

Ultimately, our focus is solely on the relationship between entropy and the
measurement. To complete the Lagrangian multiplier method, we must disregard
the intermediate dependencies $\lambda^{\prime}(r)$ outlined in equation (11).

##  IV Connecting Theory and Experiment

This is where a crucial question arises: How does the system respond to
changes in the measurements? In the experiment we can measure the changes of
$S$ on the measurements $r_{i}$. These are important quantities, so we define
this change as $\lambda_{i}$ in units of $k$:

$k\lambda_{i}(r)\coloneqq\partialderivative{S(r)}{r_{i}}$ (12)

The connection between $\lambda_{i}$ and $r_{i}$ is so fundamental that it
gets its own name. The $k\lambda_{i}$ and $r_{i}$ are called entropic
conjugated variables. To connect the experimental value eq. (12) with our
model, we calculate the derivative of eq. (11):

$\displaystyle\frac{\partial S(r)}{\partial r_{i}}$
$\displaystyle=-k\sum_{j=1}^{J}\underbrace{\frac{\partial\Omega(\lambda^{\prime})}{\partial\lambda_{j}^{\prime}}}_{r_{j}}\frac{\partial\lambda_{j}^{\prime}(r)}{\partial
r_{i}}+k\sum_{j=1}^{J}\frac{\partial\lambda_{j}^{\prime}(r)}{\partial
r_{i}}r_{j}+k\lambda_{i}^{\prime}(r)$ $\displaystyle=k\lambda_{i}^{\prime}(r)$
(13)

The combination of the experimental (exp) values eq. (12) with the model (mod)
values eq. (13) yields

$\lambda_{i}(r)\stackrel{{\scriptstyle\text{exp}}}{{=}}\frac{1}{k}\frac{\partial
S(r)}{\partial
r_{i}}\stackrel{{\scriptstyle\text{mod}}}{{=}}\lambda_{i}^{\prime}(r)$ (14)

Eq. (14) connects the experimental entropic conjugated variables
$k\lambda_{i}$ with our purely mathematical Lagrange multipliers
$k\lambda_{i}^{\prime}$. It should be emphasised that they are only equivalent
if we choose the Lagrangian multipliers as we did in eq. (8a).
With the connection to the experiment just made, we can check the meaning of
the maximum condition for the entropy for concrete systems. Since the proof
gives insight into the properties of entropy other than that it is maximal, we
will do it in detail.
To prove the maximum, one needs to compute the Hessian matrix, a matrix of
second derivatives, and check whether it is negative definite at the
measurements $r$. The Hessian matrix is calculated as:

$\frac{\partial^{2}S(r)}{\partial r_{i}\partial
r_{j}}=k\frac{\partial\lambda_{i}(r)}{\partial r_{j}}$ (15)

Despite the inability to perform the derivative of $\lambda_{i}$ with respect
to $r_{j}$ abstractly, we can employ the identity:

$\partialderivative{\lambda_{i}}{r_{j}}=\frac{1}{\partialderivative{r_{j}}{\lambda_{i}}}$
(16)

With eq. (8i), we can rewrite the derivative as:

$\frac{\partial
r_{j}(\lambda)}{\partial\lambda_{i}}=\frac{\partial^{2}\Omega(\lambda)}{\partial\lambda_{i}\partial\lambda_{j}}$
(17)

As one can see, the eigenvalues of the Hessian matrix of $\Omega$ in eq. (17)
are the inverse eigenvalues of our original Hessian matrix of $S$ in eq. (15).
This implies that it is equivalent to showing that the Hessian of
$\Omega(\lambda)$ is negative definite. Calculating the Hessian of
$\Omega(\lambda)$, we find:

$\frac{\partial^{2}\Omega(\lambda)}{\partial\lambda_{i}\partial\lambda_{j}}=\underbrace{\Tr\left(\hat{\mathrm{R}}_{i}\hat{\mathrm{\rho}}(\lambda)\right)}_{r_{i}(\lambda)}\Tr\left(\hat{\mathrm{R}}_{j}\hat{\mathrm{\rho}}(\lambda)\right)-\Tr\left(\hat{\mathrm{R}}_{i}\hat{\mathrm{R}}_{j}\hat{\mathrm{\rho}}(\lambda)\right)$
(18)

The matrix elements in eq. (18) exactly represent the negative covariance,
denoted CoV, defined as

$\\!\\!\text{CoV}_{\hat{\mathrm{\rho}}}(\hat{\mathrm{R}}_{i},\hat{\mathrm{R}}_{j})(\lambda)\coloneqq\Tr\bigg(\hat{\mathrm{R}}_{i}\hat{\mathrm{R}}_{j}\hat{\mathrm{\rho}}(\lambda)\bigg{missing})-r_{i}(\lambda)r_{j}(\lambda)$
(19)

Since the matrix elements in eq. (18) have the form of the covariance as shown
in eq. (19), it is a standard procedure to prove that the associated matrix is
negative definite. We will go through this proof for two reasons: Firstly, it
is constructive, providing insight into $\Omega(\lambda)$. Secondly, we can
extract relevant information regarding the experiment.

The Hessian of $\Omega(\lambda)$ is negative definite, i.e. $\forall
a\in\mathbb{R}^{J}:\sum\limits_{j,k=1}^{J}a_{j}\frac{\partial^{2}\Omega(\lambda)}{\partial\lambda_{j}\partial\lambda_{k}}a_{k}<0$
(20a) The proof of eq. (20a) begins by inserting the definition of the
covariance in eq. (19):
$\sum\limits_{j,k=1}^{J}\\!\\!a_{j}\frac{\partial^{2}\Omega(\lambda)}{\partial\lambda_{j}\partial\lambda_{k}}a_{k}=\\!\\!\sum\limits_{j,k=1}^{J}\\!\\!a_{j}\\!\left(\expectationvalue{\hat{\mathrm{R}}_{j}}_{\hat{\mathrm{\rho}}}\expectationvalue{\hat{\mathrm{R}}_{k}}_{\hat{\mathrm{\rho}}}-\expectationvalue{\hat{\mathrm{R}}_{j}\hat{\mathrm{R}}_{k}}_{\hat{\mathrm{\rho}}}\right)\\!a_{k}$
(20b) Next, we use the linearity of the expectation value
$=\expectationvalue{\sum\limits_{j=1}^{J}a_{j}\hat{\mathrm{R}}_{j}}_{\hat{\mathrm{\rho}}}\expectationvalue{\sum\limits_{k=1}^{J}a_{k}\hat{\mathrm{R}}_{k}}_{\hat{\mathrm{\rho}}}-\expectationvalue{\sum\limits_{j,k=1}^{J}a_{j}a_{k}\hat{\mathrm{R}}_{j}\hat{\mathrm{R}}_{k}}_{\hat{\mathrm{\rho}}}$
(20c) Rename the sum from $j$ to $k$ in the first term to arrive at
$=\expectationvalue{\sum\limits_{k=1}^{J}a_{k}\hat{\mathrm{R}}_{k}}_{\hat{\mathrm{\rho}}}^{2}-\expectationvalue{\sum\limits_{j,k=1}^{J}a_{j}a_{k}\hat{\mathrm{R}}_{j}\hat{\mathrm{R}}_{k}}_{\hat{\mathrm{\rho}}}$
(20d) Upon examining eq. (20d), one can identify the variance, indicated as
$\mathbb{V}$:
$\expectationvalue{\sum\limits_{k=1}^{J}a_{k}\hat{\mathrm{R}}_{k}}_{\hat{\mathrm{\rho}}}^{2}-\expectationvalue{\sum\limits_{j,k=1}^{J}a_{j}a_{k}\hat{\mathrm{R}}_{j}\hat{\mathrm{R}}_{k}}_{\hat{\mathrm{\rho}}}=-\mathbb{V}_{\hat{\mathrm{\rho}}}\left(\sum\limits_{k=1}^{J}a_{k}\hat{\mathrm{R}}_{k}\right)$
(20e) The variance is positive, which means that eq. (20e) is always less than
zero for any $a$, which is exactly what we wanted.

We have just proved that the Hessian of $\Omega(\lambda)$ is negative definite
independent of $\lambda$, which implies that the entropy $S(r)$ has the same
properties independent of $r$. From a mathematical point of view, we can
directly derive the so-called stability conditions for the experiment. We know
that the $i^{\text{th}}$ diagonal element of the Hessian eq. (18) are the
negative variance of $\hat{\mathrm{R}}_{i}$ . From that, we obtain the one-
parameter stability condition as

$\partialderivative{r_{i}}{\lambda_{i}}<0$ (21)

For the experiment eq. (21) tells us that changing one parameter $\lambda_{i}$
must change the associated $r_{i}$ with a negative slope, like a self-
stabilising system. From mathematics we know that if a matrix is negative
definite, then the determinants of all submatrices must also be negative. For
changing two parameters in the experiment we can then expect the two-parameter
stability conditions:

$\displaystyle\det(\begin{matrix}\partialderivative{r_{i}}{\lambda_{i}}&\partialderivative{r_{j}}{\lambda_{i}}\\\
\partialderivative{r_{i}}{\lambda_{j}}&\partialderivative{r_{j}}{\lambda_{j}}\end{matrix})<0\Rightarrow\partialderivative{r_{i}}{\lambda_{i}}\partialderivative{r_{j}}{\lambda_{j}}<\partialderivative{r_{i}}{\lambda_{j}}\partialderivative{r_{j}}{\lambda_{i}}$
(22)

This principle can be extended to include all $J$ parameters in the
experiment.
It should be noted that measuring the change
$\partialderivative{r_{j}}{\lambda_{i}}$ may be experimentally difficult or
even impossible. However, it can often be determined by using eq. (17):

$\frac{\partial
r_{j}(\lambda)}{\partial\lambda_{i}}=\frac{\partial^{2}\Omega(\lambda)}{\partial\lambda_{i}\partial\lambda_{j}}=\partialderivative{r_{i}(\lambda)}{\lambda_{j}}$
(23)

The order of the partial derivatives is of no concern, so it is easy to
construct the so-called Maxwell-relations eq. (23) which are heavily used in
experiments.
Above we proved that the Hessian of $S(r)$ is negative for every $r$, meaning
that the entropy is even concave. This is a powerful property and will be used
to extend the possibilities of modeling the experiments by including
reservoirs. The resulting technique we need is the Legendre transformation.

##  V Legendre Transformation

From an experimental perspective, it can be advantageous to adjust the
associated conjugate quantity $\lambda_{i}$ instead of measuring certain
quantities $r_{i}$. In statistical physics, this means describing the entropy
in terms of $\lambda_{i}$ rather than $r_{i}$. This approach allows us to
describe the function $S(r)$ with its slope
$\lambda_{i}(r)\stackrel{{\scriptstyle\text{exp}}}{{=}}\frac{1}{k}\frac{\partial
S(r)}{\partial r_{i}}$. The technical process for achieving this is outlined
in fig. 2, there is an example of a quadratic function.

Figure 2: The function $y=f(x)$ is plotted based on $x$. An arbitrary point
$x_{0}$ is selected. The slope $s$ of the function $f$ at point $x_{0}$ is
given as $s=f^{\prime}(x_{0})$ and the offset $d$ of the resulting tangent
$y=s\,x+d$ at point $(x_{0},y_{0})$ is given by $d=y-s\,x$.

The diagram demonstrates how to alter the dependence without losing
information. To encode the information of the position $x_{0}$ and the
associated value $y_{0}$ in a function $f^{*}(s)$, we require a second
characteristic of the tangent: the offset of the tangent $d$, which is
dependent on the slope $s$. Geometrically, there are two ways to convert the
full information of $f(x)$:

$\displaystyle\pm f^{*}(s)=d(x(s))=y-s\,x=f(x(s))-s\,x(s)$ (24)

In eq. (24), we can see that the describing function resulting from using the
slope $s$ instead of $x$ is the slope dependence offset $d(s)$. The question
that naturally arises is: what are the necessary conditions for this type of
transformation to be unique? Fortunately, mathematicians have answered this
question: $f^{*}(s)$ is only a function, with a unique $d$ value for each $s$,
if $f$ is convex. To gain an understanding of convex functions, we will
demonstrate the concept geometrically using the function $f(x)=\exp(x)$ in
fig. 3.

Figure 3: The plot displays the function $f(x)=\exp(x)$ with two points:
$(x_{1},\exp(x_{1}))$ and $\left(x_{2},\exp\left(x_{2}\right)\right)$. A
dashed (blue) line connects these points.

There, it is shown that the line $y(x)$ between the two points
$(x_{1},\exp(x_{1}))$ and $(x_{2},\exp(x_{2}))$ is parameterized as:

$y(t)=\exp(x_{2})t+\exp(x_{1})(1-t)$ (25)

with $t=\frac{x-x_{1}}{x_{2}-x_{1}}$. Therefore, for $t\in[0,1]$ the line is
restricted between the points $(x_{1},\exp(x_{1}))$ and $(x_{2},\exp(x_{2}))$
and it is always greater than $f(x)=\exp(x)$ for $x_{1}\leq x\leq x_{2}$. This
property defines a convex function. Convexity allows us to properly define the
transformation in eq. (24):

Definition V.1:  Legendre transformation Let
$f:\mathbb{R}^{J}\rightarrow\mathbb{R}$ be a convex function, i.e. $\forall
t\in[0,1]:\forall r,s\in\mathbb{R}^{J}:$ $f(t\cdot r+(1-t)\cdot s)\leq
tf(r)+(1-t)f(s)$ Then the Legendre transformation (LT) of $f$ with
$r_{i}\rightarrow\lambda_{i}$, written as
$f_{i}^{*}(r_{1},\dots,\lambda_{i},\dots,r_{j})$ is defined as
$f_{i}^{*}(r_{1},\dots,\lambda_{i},\dots,r_{j})\coloneqq\sup_{r_{i}\in\mathbb{R}}(r_{i}\lambda_{i}-f(r))$
(26)

The LT possesses significant mathematical properties, which we will reiterate
by utilizing the results on our entropy. These properties are crucial:

  * •

The function $f^{*}$ is once again convex. This property is encountered in
thermodynamics and classical mechanics. When starting from a convex potential,
the LT inherits this property. In application, if there is an energy minimum,
the transformed energy is also realized as a minimum.

  * •

$f$ and $f^{*}$ are linked symmetrically:

$\forall r,\lambda\in\mathbb{R}:\,f(r)+f^{*}(\lambda)\geq r\lambda$ (27)

The equality in eq. (27) holds when $r$ maximises the supremum in eq. (26), we
will denote this by $r(\lambda)$. This property will be crucial in the
discussion of thermodynamics.

  * •

The LT is an involution, therefore

$(f^{*})^{*}(r)=\sup_{\lambda\in\mathbb{R}}(r\lambda-f^{*}(\lambda))=f(r)$
(28)

The complete information has been preserved as desired.

  * •

If $f$ is differentiable we can calculate the maximum of $r\lambda-f(r)$ and
find $r(\lambda)$. This results in the following equation:

$\displaystyle\lambda=f^{\prime}(r(\lambda))$ (29)

The unique solution of this equation is $r(\lambda)$, so we get for the LT of
$f$

$f^{*}(\lambda)=r(\lambda)f^{\prime}(r(\lambda))-f(r(\lambda))$ (30)

Although the LT is typically defined for convex functions, this concept can
also be extended to concave functions such as entropy. In our case, the
entropy $S(r)$ is differentiable, leading to the following definition

Definition V.2:  Concave A function $g$ is considered concave if its negative,
$-g$, is convex. The LT of a concave function $g$ is defined as
$g^{*}(\lambda)\coloneqq-r(\lambda)g^{\prime}(r(\lambda))+g(r(\lambda))$ (31)

It is important to note that while the entropy $S(r)$ is concave,
$S^{*}(\lambda)$ is convex and $(S^{*})^{*}(r)=S(r)$ is concave again. This
highlights the difference between whether the function is concave or convex,
which must also be considered in its application: In mechanics, the kinetic
energy is a convex function, To switch between the Lagrange and Hamiltonian
formalism, we use eq. (30). However, in statistical physics, we deal with the
concave entropy. By applying eq. (31) to our concave entropy $S$, we have
already calculated the necessary component in eq. (14):

$\frac{\partial S(r)}{\partial r_{i}}=k\lambda_{i}$

As previously mentioned, $r_{i}$ and $k\lambda_{i}$ have a special
relationship as entropy-conjugated variables. This relationship also reappears
in the LT. Therefore it is worth introducing the notation for entropy-
conjugated variables:

$r_{i}\stackrel{{\scriptstyle S}}{{\Leftrightarrow}}k\lambda_{i}$

To clarify the concept of LT and entropy-conjugated variables, let us
demonstrate using the most relevant experimental example:
LT of $S$ concerning only the energy:
Assuming $J=1$ and the only observable is
$\hat{\mathrm{R}}_{1}=\hat{\mathrm{H}}$, the Hamiltonian with energy $E$ as
the measurement value $r_{1}=E$. Then the entropy $S$ can be calculated using
eq. (11)

$S(E)=-k\Omega(\lambda(E))+k\lambda(E)E$ (32)

Suppose the measured value $E$ is adjusted using a reservoir. A reservoir
refers to a system larger than the one considered and acts as a buffer, thus
operating as a sink or source of entropy. We want to include the exchange with
the reservoir in the entropy calculation, resulting in the combined entropy
denoted as $S^{*}$. To calculate this, we use the LT and require the entropy-
conjugated variable of $E$, which is defined from a measurement as

$\beta(E)\equiv\lambda(E)\stackrel{{\scriptstyle\text{exp}}}{{=}}\frac{1}{k}\frac{\partial
S(E)}{\partial E}$ (33)

with $\beta$ as the coldness, which is related to the temperature $T$ via

$T=\frac{1}{k\beta}$ (34)

assuming $k$ is fixed to the Boltzmann-Constant. The coldness $\beta$
parameterises the entropy exchange between the system we want to study and the
reservoir. To calculate $S^{*}$, we can use the experimental knowledge
$E\stackrel{{\scriptstyle S}}{{\Leftrightarrow}}k\beta$ and eq. (31):

$S^{*}(\beta)=S(E(\beta))-k\beta E(\beta)$ (35a) The first part of eq. (35a)
represents the entropy of the original system, now depending on $\beta$, while
the second part represents the exchanged entropy with the reservoir. By
inserting eq. (11) into eq. (35a) we find that
$S^{*}(\beta)=-k\Omega(\underbrace{\beta(E(\beta))}_{\beta})\overbrace{+k\underbrace{\beta(E(\beta))}_{\beta}E(\beta)-k\beta
E(\beta)}^{0}$ (35b) which can be simplified to $S^{*}(\beta)=-k\Omega(\beta)$
(35c)

As shown in eq. (35c), the fully Legendre transformed $S^{*}$ can be
calculated directly by knowing $\Omega$ from eq. (10). This means that the
entropy including all reservoirs is purely determined by the coupling
parameter namely the coldness $\beta$. This concept can be easily extended to
a system with any number of measurements including their respective
reservoirs:

$S^{*}(\lambda)=-k\Omega(\lambda)$ (36)

In this discussed case, with only one reservoir having the coupling $\beta$,
the combined entropy $S^{*}(\beta)$ is referred to as the free entropy and is
linked to the free energy $F$ through

$F(\beta)\coloneqq-\frac{S^{*}(\beta)}{k\beta}=\frac{1}{\beta}\Omega(\beta)=-\frac{1}{\beta}\ln\Bigg(\Tr\bigg(\exp(-\beta\hat{\mathrm{H}})\bigg{missing})\Bigg{missing})$
(37)

In cases where $J>1$, there are various options for LTs depending on the
experimental setup, and the quantities of interest, as shown in fig. 4.

$S_{12}^{*}(\lambda_{1},\lambda_{2},r_{3})$$S(r_{1},r_{2},r_{3})$$S(\lambda_{1},r_{2},r_{3})$$S(r_{1},\lambda_{2},r_{3})$$S(\lambda_{1},\lambda_{2},r_{3})$$S_{1}^{*}(\lambda_{1},r_{2},r_{3})$$k\lambda_{2}r_{2}$$k\lambda_{1}r_{1}$$S_{2}^{*}(r_{1},\lambda_{2},r_{3})$

Figure 4: This is an illustration of the experimental setup of a system with
two reservoirs. The system includes three measurements: $r_{1}$, $r_{2}$, and
$r_{3}$ and contains the entropy $S$. The combined entropy of the system and
reservoir 1 is denoted as $S_{1}^{*}$ and the combination with reservoir 2 is
denoted as $S_{2}^{*}$. The total entropy including both reservoirs 1 and 2 is
denoted as $S_{12}^{*}$.

This figure allows us to provide a physical interpretation for the
mathematically geometric LT in the general case where $J>1$. The LT expands
our entropy to include the selected reservoir:
$S_{1}^{*}(\lambda_{1},r_{2},r_{3})$ represent the entropy from our system,
including reservoir 1, through the coupling $k\lambda_{1}$:

$\displaystyle
S_{1}^{*}(\lambda_{1},r_{2},r_{3})=S(\lambda_{1},r_{2},r_{3})-k\lambda_{1}r_{1}(\lambda_{1},r_{2},r_{3})$
(38a) The first part of eq. (38a) represents the entropy of the original
system, and the second represents the exchanged entropy with reservoir 1. The
repetition of the same LT of $S_{1}^{*}(\lambda_{1},r_{2},r_{3})$ eq. (38a)
results in the removal of reservoir 1:
$\displaystyle\left(S_{1}^{*}\right)_{1}^{*}(r_{1},r_{2},r_{3})$
$\displaystyle=S_{1}^{*}(r_{1},r_{2},r_{3})+k\lambda_{1}(r_{1},r_{2},r_{3})r_{1}$
$\displaystyle=S(r_{1},r_{2},r_{3})$
$\displaystyle\underbrace{-k\lambda_{1}(r_{1},r_{2},r_{3})r_{1}+k\lambda_{1}(r_{1},r_{2},r_{3})r_{1}}_{=0}$
(38b) An alternative approach is to incorporate reservoir 2 by means of LT eq.
(38a), using the coupling $k\lambda_{2}$: $\displaystyle
S_{12}^{*}(\lambda_{1},\lambda_{2},r_{3})=S_{1}^{*}(\lambda_{1},\lambda_{2},r_{3})-k\lambda_{2}r_{2}(\lambda_{1},\lambda_{2},r_{3})$
(38c) The first part of eq. (38c) describes the entropy of the system
including the coupling to reservoir 1 and the second part the entropy
exchanged with the reservoir 2. To exclude reservoir 1 or/and 2, LT can be
applied again. The same discussion as in fig. 4 can also be done by starting
LT to include reservoir 2, leading to the same result.

It is important to note that all properties in fig. 4 can be calculated if
$S(r_{1},r_{2},r_{3})$, $S_{1}^{*}(\lambda_{1},r_{2},r_{3})$,
$S_{2}^{*}(r_{1},\lambda_{2},r_{3})$ or
$S_{12}^{*}(\lambda_{1},\lambda_{2},r_{3})$ is known. It should be noted that
although they have different experimental meanings, they are connected via the
LT.
In practice, dealing with such systems involves first calculating the combined
entropy eq. (36) including all reservoirs. To obtain the desired information,
one can then start Legendre transforming $S^{*}(\lambda)$. Although
$S^{*}(\lambda)$ is convex by construction, it is necessary to use the LT eq.
(31) because the original entropy $S$ is concave. This formalism allows for
the calculation of the entropy of the original system and the exchange entropy
with the associated reservoir.

##  VI Full Formalisation of the Experimental Setup

Experimental setups can become complex. To control this complexity, we
formalise the concept shown in fig. 4. To determine the number and type of
observables considered in the experiment, an ensemble is introduced:

Definition VI.1:  Statistical Ensembles A statistical ensemble is a list of
observables $\big{(}\\{\hat{\mathrm{R}}_{j}\\}_{j=1}^{J}\big{)}$, denoted as
($\hat{\mathrm{R}}_{1},\hat{\mathrm{R}}_{2},\dots,\hat{\mathrm{R}}_{J}$).

To represent such an ensemble we have already utilised

$C_{r}\coloneqq r=(r_{1},r_{2},\dots,r_{J})$
$C_{\lambda}\coloneqq\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{J})$

giving us the familiar entropy that depends on $r$ or $\lambda$

$S(C_{r})=S(r)\text{ or }S(C_{\lambda})=S(\lambda)$

To generalize this concept, one must choose either $r_{i}$ or $k\lambda_{i}$
for each reservoir presented in the experimental setup. It is not possible to
have a dependency of both since they are entropy-conjugated
$r_{i}\stackrel{{\scriptstyle S}}{{\Leftrightarrow}}k\lambda_{i}$. This choice
between $r_{i}$ and $k\lambda_{i}$ is called a representation of the
statistical ensemble
($\hat{\mathrm{R}}_{1},\hat{\mathrm{R}}_{2},\dots,\hat{\mathrm{R}}_{J}$):

Definition VI.2:  Representation of a Statistical Ensembles Consider a
statistical ensemble denoted by
($\hat{\mathrm{R}}_{1},\hat{\mathrm{R}}_{2},\dots,\hat{\mathrm{R}}_{J}$),
where the exclusive choice between $r_{i}$ and its entropy-conjugated quantity
$k\lambda_{i}$ is made for every present reservoir $k\lambda_{i}r_{i}$. If no
reservoir is present, $r_{i}$ is chosen, resulting in a representation $C$ of
the statistical ensemble.

Examples of statistical ensembles and their representations are:
$(\hat{\mathrm{H}})$ $\Rightarrow$ $(E)$ $\Leftrightarrow$ $(\beta)$
$(\hat{\mathrm{N}})$ $\Rightarrow$ $(\expectationvalue{\hat{\mathrm{N}}})$
$\Leftrightarrow$ $(\beta\mu)$ $(\hat{\mathrm{H}},\hat{\mathrm{N}})$
$\Rightarrow$ $(E,\expectationvalue{\hat{\mathrm{N}}})$ $\Leftrightarrow$
$(\beta,\expectationvalue{\hat{\mathrm{N}}})$ $\Leftrightarrow$ $(E,\beta\mu)$
$\Leftrightarrow(\beta,\beta\mu)$
Consider an ensemble
$(\hat{\mathrm{R}}_{1},\hat{\mathrm{R}}_{2},\hat{\mathrm{R}}_{3},\hat{\mathrm{R}}_{4})$
with two reservoirs having coupling parameters $k\lambda_{2}$ and
$k\lambda_{3}$. The representations $C$ of this ensemble are as follows:

$\displaystyle C_{r}$ $\displaystyle=(r_{1},r_{2},r_{3},r_{4})$ $\displaystyle
C_{1}$ $\displaystyle=(r_{1},r_{2},\lambda_{3},r_{4})$ $\displaystyle C_{2}$
$\displaystyle=(r_{1},\lambda_{2},r_{3},r_{4})$ $\displaystyle C_{3}$
$\displaystyle=(r_{1},\lambda_{2},\lambda_{3},r_{4})$

As outlined in the definition of representations $C$, $r_{1}$ and $r_{4}$ must
be selected, as there is no coupling to their respective reservoirs.
For any statistical ensemble and a given representation $C$, it is possible to
calculate the entropy of the original system as

$S(C)=-k\Omega(C)+k\sum_{\lambda_{i}\in C}\lambda_{i}r_{i}(C)+k\sum_{r_{i}\in
C}\lambda_{i}(C)r_{i}$ (39)

where

$\Omega(C)=-\ln\\!\left(\\!\Tr\\!\left(\\!\exp\\!\\!\Bigg{(}\\!\\!-\\!\\!\\!\sum_{\lambda_{i}\in
C}\lambda_{i}\hat{\mathrm{R}}_{i}-\sum_{r_{i}\in
C}\lambda_{i}(C)\hat{\mathrm{R}}_{i}\\!\\!\Bigg{)}\\!\\!\right)\\!\\!\right)$
(40)

As previously stated, this is only a portion of the total entropy of the
system. In order to calculate a combined entropy $S_{C^{\prime}}^{*}(C)$, one
must first select a representation $C$ and a list of coupling parameters
$\lambda_{i}$ for the chosen reservoirs, which are encoded in the
representation $C^{\prime}$:

$\displaystyle
S_{C^{\prime}}^{*}(C)=-k\Omega(C)+k\sum_{\begin{subarray}{c}\lambda_{i}\in
C\\\ \ \lambda_{i}\notin
C^{\prime}\end{subarray}}\lambda_{i}r_{i}(C)+k\sum_{\begin{subarray}{c}r_{i}\in
C\\\ \ r_{i}\in C^{\prime}\end{subarray}}\lambda_{i}(C)r_{i}$ (41)

Statistical potentials can be derived from eq. (41), by setting
$C=C^{\prime}$.

Definition VI.3:  Statistical Potentials A statistical potential is the LT,
according to $C^{\prime}$, of $S(r)$ with the representation $C=C^{\prime}$ of
the statistical ensemble
($\hat{\mathrm{R}}_{1},\hat{\mathrm{R}}_{2},\dots,\hat{\mathrm{R}}_{J}$):
$S_{C}^{*}(C)=-k\Omega(C)+k\sum_{r_{i}\in C}\lambda_{i}(C)r_{i}$ (42)

Potentials and their associated representations are described using natural
variables ($C=C^{\prime}$). Although eq. (41) appears more general, only the
statistical potentials eq. (42) contain the full information on the
experimental setup. It is only from these statistical potentials that every
other entropy eq. (41) can be calculated.
Finally, there is one question left to answer: ’What does this have to do with
thermodynamics?’

##  VII From Statistical Physics to Thermodynamics

To derive thermodynamics (TD) from statistical physics, it is shown how to get
the axioms of thermodynamics [2]. In TD, the primary quantity of interest is
energy $E$. To provide a historical perspective, the focus must shift from
entropy to energy. To accomplish this, we consider that in thermodynamics, the
energy observable $\hat{\mathrm{H}}$ is always present. Without loss of
generality, we select $\hat{\mathrm{R}}_{1}=\hat{\mathrm{H}}$, yielding
$r_{1}=E$. Its entropy-conjugated quantity $E\stackrel{{\scriptstyle
S}}{{\Leftrightarrow}}k\beta$ corresponds to the coldness $\lambda_{1}=\beta$.
By inserting this information into eq. (11), we find

$S(E,r)=-k\Omega(E,r)+k\beta(E,r)E+k\sum_{j=2}^{J}\lambda_{j}(E,r)r_{j}$ (43a)
where $r=(r_{2},\dots,r_{j})$. To express the entropy-conjugated quantities
through energy-conjugated ones, we need to alter the point of view from
entropy $S(E,r)$ to energy $E(S,r)$. This requires using the chain rule on the
entropy-conjugated quantities: $k\lambda_{j}=\frac{\partial S}{\partial
r_{j}}=\frac{\partial S}{\partial E}\frac{\partial E}{\partial
r_{j}}=k\beta\frac{\partial E}{\partial r_{j}}$ (43b) Using the chain rule eq.
(43b), we can define the energy-conjugated variables as
$r_{i}\stackrel{{\scriptstyle E}}{{\Leftrightarrow}}k\Lambda_{i}$ (43c) with
$\Lambda_{i}\coloneqq\frac{\partial E}{\partial r_{i}}$ (43d) Upon examining
eq. (43b) and eq. (43d), it becomes apparent that the variables conjugated to
entropy and energy are linked through the variable $\beta$, specifically
$\lambda_{i}=\beta\Lambda_{i}$ (43e) Using identity eq. (43e), we can express
entropy eq. (43a) and eq. (10) in terms of energy-conjugated variables:
$\begin{split}S(E,r)=&-k\Omega(E,r)+k\beta(E,r)E\\\
&+k\beta(E,r)\sum_{j=2}^{J}\Lambda_{j}(E,r)\,r_{j}\end{split}$ (43f)
$\Omega(E,r)=-\ln\\!\left(\\!\Tr\\!\left(\\!\exp\\!\\!\Bigg{(}\\!\\!-\beta(E,r)\\!\bigg{(}\hat{\mathrm{H}}+\\!\\!\sum_{j=2}^{J}\Lambda_{j}(E,r)\hat{\mathrm{R}}_{j}\bigg{)}\\!\\!\Bigg{)}\\!\\!\right)\\!\\!\right)$
(43g)

Some examples of conjugated variables are

$E\stackrel{{\scriptstyle S}}{{\Leftrightarrow}}k\beta$, |  |  $S\stackrel{{\scriptstyle E}}{{\Leftrightarrow}}\frac{1}{k\beta}$, |  |  $N\stackrel{{\scriptstyle E}}{{\Leftrightarrow}}\mu$, |  | $X\stackrel{{\scriptstyle E}}{{\Leftrightarrow}}\frac{\partial E}{\partial X}$
---|---|---|---|---|---|---

From a statistical ensemble, which includes the Hamiltonian
($\hat{\mathrm{H}},\\{\hat{\mathrm{R}}_{j}\\}_{j=2}^{J}$), we can construct a
TD ensemble. This TD ensemble differs only in the representation. One must
choose between $r_{i}$ and $\Lambda_{i}$. In TD, we mainly deal with three
ensembles: the micro-canonical, canonical, and grand canonical ensembles. In
the microcanonical ensemble, the probabilities of the state are equally
distributed as there are no measurements. In the canonical ensemble, the only
observable is $\hat{\mathrm{H}}$, resulting in the Boltzmann distribution. In
the grand canonical ensemble, there are two observables: the Hamiltonian
$\hat{\mathrm{H}}$ and the number of particles $\hat{\mathrm{N}}$. A
Hamiltonian $\hat{\mathrm{H}}$ that is linear in $\hat{\mathrm{N}}$ leads to
the Bose-Einstein distribution for bosons and the Fermi-Dirac distribution for
fermions. When calculating these ensembles, TD potentials are also introduced.
By applying the chain rule to the same conversion procedure, one can find

$U_{C}^{*}(C)\coloneqq-\frac{S_{C}^{*}(C)}{k\beta(C)}$ (44)

The TD potential in eq. (44) is represented by $U_{C}^{*}(C)$. An example of
such a potential has already been encountered in eq. (37), where the potential
was referred to as the free energy. If no reservoirs are present, meaning the
only representation is $C=C_{r}$, the TD potential is then called the inner
energy $U$, which is defined as

$U(r)\coloneqq U_{C_{r}}^{*}(C_{r}).$ (45)

The inner energy serves as the starting potential for every thermodynamic
analysis. All relevant quantities can be extracted from the TD potential by
differentiation. If the coldness $\beta$ is part of the representation $C$ the
entropy can be recovered, by using eqs. (44) and (42)

$\partialderivative{U_{C}^{*}(C)}{\beta}=-\frac{1}{\beta^{2}}\Omega(C)+\frac{1}{\beta}\partialderivative{\Omega(C)}{\beta}-\sum\limits_{r_{i}\in
C}\partialderivative{\Lambda_{i}(C)}{\beta}r_{i}$ (46a) where
$\Omega(C)=-\ln\\!\left(\\!\Tr\\!\left(\\!\exp\\!\\!\Bigg{(}\\!\\!-\\!\\!\\!\beta\bigg{(}\hat{\mathrm{H}}+\sum_{\Lambda_{i}\in
C}\Lambda_{i}\hat{\mathrm{R}}_{i}+\sum_{r_{i}\in
C}\Lambda_{i}(C)\hat{\mathrm{R}}_{i}\bigg{)}\\!\\!\Bigg{)}\\!\\!\right)\\!\\!\right)$
(46b) The second term in eq. (46a) is calculated as
$\displaystyle\partialderivative{\Omega(C)}{\beta}=$ $\displaystyle
E(C)+\sum_{\Lambda_{i}\in C}\Lambda_{i}r_{i}(C)+\sum_{r_{i}\in
C}\Lambda_{i}(C)r_{i}$ $\displaystyle+\beta\sum_{r_{i}\in
C}\partialderivative{\Lambda_{i}(C)}{\beta}r_{i}$ (46c) Inserting eq. (46c)
into eq. (46a) one finds
$\displaystyle\partialderivative{U_{C}^{*}(C)}{\beta}=$
$\displaystyle-\frac{1}{\beta^{2}}\Omega(C)+\frac{1}{\beta}E(C)+\frac{1}{\beta}\sum_{\Lambda_{i}\in
C}\Lambda_{i}r_{i}(C)$ $\displaystyle+\frac{1}{\beta}\sum_{r_{i}\in
C}\Lambda_{i}(C)r_{i}$ (46d) By using eq. (39) and eq. (43e), eq. (46d) can be
simplified to obtain the final result
$\partialderivative{U_{C}^{*}(C)}{\beta}=\frac{1}{k\beta^{2}}S(C)$ (46e)

Eq. (46e) shows that starting from the TD potential allows for the
reconstruction of entropy, which is a common practice in thermodynamics. The
properties of the system are calculated based on the potential. In TD,
temperature $T$ is typically used instead of coldness $\beta$. By applying the
chain rule to eq. (46e) with respect to $\beta$, we arrive at a fundamental
result in TD:

$\partialderivative{U_{C}^{*}(C)}{T}=-S(C)$ (47)

In the context of phenomenological TD, temperature is considered a fundamental
property. By measuring the temperature-dependent potential, one can directly
observe the entropy as the slope of the potential eq. (47).
To conclude the discussion on TD, we will establish the connection to the
axioms of phenomenological TD. These axioms include the Euler inequality and
the state function, which we will prove using statistical physics

Axiom :  State function There exists a convex function $\Delta(\beta,\Lambda)$
dependent solely on energy-conjugate variables $\beta$ and $\Lambda$, termed
as the state function, which is monotonically increasing in $\beta$ and
monotonically increasing (decreasing) in $\Lambda_{i}$ if $r_{i}<0\,(>0)$. The
energy-conjugate quantities must satisfy the equilibrium condition
$\Delta(\beta,\Lambda)=0$ (48) The set of $\beta$ and $\Lambda$ values
satisfying the equilibrium condition eq. (48) is referred to as the state
space.

Phenomenological thermodynamics begins with the equation of state (48). This
function is constructed using experimental data, more fundamental theories
such as statistical physics, or models. In statistical physics, the state
function is represented by

$\Delta(\beta,\Lambda)\coloneqq-\frac{\Omega(\beta,\Lambda)}{\beta}$ (49)

Eq. (37) previously encountered a state function of this kind.
The proof of convexity, which satisfies the axiom from above, is given in
Appendix B. The second axiom of TD discusses the Euler inequality.

Axiom :  Euler inequality The energy $E$ satisfies the Euler inequality $E\geq
TS-\sum_{j=2}^{J}r_{i}\Lambda_{i}$ (50) for all $T$ and $\Lambda$ in the state
space. In equilibrium equality holds.

From eq. (27) and the concavity of $S$, it can be observed that
$S_{C_{\lambda}}^{*}(\beta,\lambda)-S(E,r)\geq-k\beta
E-k\sum_{j=2}^{J}r_{j}\lambda_{j}$ (51a) Using eq. (36) $k\beta E\geq
k\Omega(\beta,\Lambda)+S(E,r)-k\sum_{j=2}^{J}r_{j}\lambda_{j}$ (51b) and by
dividing both sides by $k\beta$ and utilizing eq. (43e) we obtain
$E\geq\frac{\Omega(\beta,\Lambda)}{\beta}+\frac{S(E,r)}{k\beta}-\sum_{j=2}^{J}r_{j}\Lambda_{j}$
(51c) Using eq. (49) one arrives at
$E\geq-\Delta(\beta,\Lambda)+\frac{S(E,r)}{k\beta}-\sum_{j=2}^{J}r_{j}\Lambda_{j}$
(51d) Finally, by restricting $\beta$ and $\Lambda$ to the state space eq.
(48) and expressing $\beta$ in terms of $T$ we arrive at the Euler inequality:
$E\geq TS(E,r)-\sum_{j=2}^{J}r_{j}\Lambda_{j}$ (51e)

Eqs. (48) and (50) show that TD focuses on equilibrium conditions, while
statistical physics encompasses both equilibrium and non-equilibrium physics.
However, this generality is accompanied by the disadvantage of not utilising
problem-specific properties. In the context of specific problems, the choice
between the methods of phenomenological TD and statistical physics may be
made.

##  VIII Conclusion

This paper serves as a pedestrian guide to understanding the three fundamental
properties of entropy, namely, the additive property of independent
subsystems, the maximisation of uncertainty in unknown systems, and the
absence of entropy in known systems. It elucidates these properties through
illustrative examples. This work demonstrates how entropy can be determined
from experimental data by making the physical intuition mathematically
precise, leading to stability conditions and Maxwell relations. Furthermore,
the paper elucidates the interplay between theory and experiment. It
illustrates how experimental manipulation via reservoirs influences theory via
the Legendre transformation, which was explored geometrically, mathematically
and physically. This paper provides a precise definition of ensembles with
their representation, highlighting the different entropy functions.
Additionally, it offers a definition of potentials with their associated
natural variables and insights into how they can be used effectively in
statistical physics. Finally, the paper concludes by deriving the theoretical
frameworks of phenomenological thermodynamics, which consist of the state
function, their convexity and the Euler inequality, through statistical
physics.

###### Acknowledgements.

We express our gratitude to Thomas Winkler and Dr. Kurt Hingerl for the
stimulating discussions and critical reading of the manuscript.

##  Appendix A Entropy’s of Independent Subsystems are Additive

Assuming that the state $\hat{\mathrm{\rho}}_{\text{AB}}$ comprises of two
independent subsystems $\hat{\mathrm{\rho}}_{\text{A}}$ and
$\hat{\mathrm{\rho}}_{\text{B}}$:

$\hat{\mathrm{\rho}}_{\text{AB}}=\hat{\mathrm{\rho}}_{\text{A}}\otimes\hat{\mathrm{\rho}}_{\text{B}}$
(52)

The entropy of the entire system can be calculated using

$S[\hat{\mathrm{\rho}}_{\text{AB}}]=-k\Tr_{\text{AB}}\left(\hat{\mathrm{\rho}}_{\text{AB}}\ln\left(\hat{\mathrm{\rho}}_{\text{AB}}\right)\right)$
(53)

By inserting the two independent subsystems eq. (52) one can arrive at

$\Tr_{\text{AB}}\left(\hat{\mathrm{\rho}}_{\text{AB}}\ln\left(\hat{\mathrm{\rho}}_{\text{AB}}\right)\right)=\Tr_{\text{AB}}\left(\hat{\mathrm{\rho}}_{\text{A}}\otimes\hat{\mathrm{\rho}}_{\text{B}}\ln\left(\hat{\mathrm{\rho}}_{\text{A}}\otimes\hat{\mathrm{\rho}}_{\text{B}}\right)\right)$
(54)

As systems A and B are independent, it is possible to decompose the tensor
product into a sum of logarithms:

$\ln\left(\hat{\mathrm{\rho}}_{\text{A}}\otimes\hat{\mathrm{\rho}}_{\text{B}}\right)=\ln\left(\hat{\mathrm{\rho}}_{\text{A}}\right)+\ln\left(\hat{\mathrm{\rho}}_{\text{B}}\right)$
(55)

and find

$\displaystyle\Tr_{\text{AB}}\left(\hat{\mathrm{\rho}}_{\text{A}}\otimes\hat{\mathrm{\rho}}_{\text{B}}\ln\left(\hat{\mathrm{\rho}}_{\text{A}}\otimes\hat{\mathrm{\rho}}_{\text{B}}\right)\right)=$
$\displaystyle\Tr_{\text{AB}}\left(\hat{\mathrm{\rho}}_{\text{A}}\otimes\hat{\mathrm{\rho}}_{\text{B}}\left(\ln\left(\hat{\mathrm{\rho}}_{\text{A}}\right)+\ln\left(\hat{\mathrm{\rho}}_{\text{B}}\right)\right)\right)$
(56)

By applying the definition of evaluating the trace $\Tr_{\text{AB}}$ as

$\Tr_{\text{AB}}(\hat{\mathrm{\rho}})\coloneqq\Tr_{\text{A}}\left(\Tr_{\text{B}}\left(\hat{\mathrm{\rho}}\right)\right)=\Tr_{\text{B}}\left(\Tr_{\text{A}}\left(\hat{\mathrm{\rho}}\right)\right)$
(57)

eq. (56) can be simplified to

$\displaystyle\Tr_{\text{A}}\left(\hat{\mathrm{\rho}}_{\text{A}}\ln\left(\hat{\mathrm{\rho}}_{\text{A}}\right)\right)+\Tr_{\text{B}}\left(\hat{\mathrm{\rho}}_{\text{B}}\ln\left(\hat{\mathrm{\rho}}_{\text{B}}\right)\right)$
(58)

These terms represent the entropy of the two subsystems.

$-k\Tr_{\text{A}}\left(\hat{\mathrm{\rho}}_{\text{A}}\ln\left(\hat{\mathrm{\rho}}_{\text{A}}\right)\right)-k\Tr_{\text{B}}\left(\hat{\mathrm{\rho}}_{\text{B}}\ln\left(\hat{\mathrm{\rho}}_{\text{B}}\right)\right)=S[\hat{\mathrm{\rho}}_{\text{A}}]+S[\hat{\mathrm{\rho}}_{\text{B}}]$
(59)

We have proven that the entropy of independent subsystems satisfy

$\displaystyle
S[\hat{\mathrm{\rho}}_{\text{AB}}]=S[\hat{\mathrm{\rho}}_{\text{A}}]+S[\hat{\mathrm{\rho}}_{\text{B}}]$
(60)

as desired.

##  Appendix B $\Delta$ is Convex

To prove $\Delta$ is convex, it is necessary to demonstrate that the Hessian
is positive definite. To obtain the Hessian, we require the second derivatives
of $\Delta$. For this, we calculate the first derivatives of $\Delta$

$\displaystyle\partial_{\beta}\Delta$
$\displaystyle=\frac{1}{\beta^{2}}\Omega-\frac{1}{\beta}\partial_{1}\Omega-\frac{1}{\beta}\sum_{j=2}^{J}\Lambda_{j}\partial_{j}\Omega$
$\displaystyle\partial_{\Lambda_{j}}\Delta$
$\displaystyle=-\partial_{j}\Omega$ (61)

With this, we can proceed to the second derivatives

$\displaystyle\partial^{2}_{\beta}\Delta$
$\displaystyle=-\frac{2}{\beta^{3}}\Omega+\frac{2}{\beta^{2}}\partial_{1}\Omega+\frac{2}{\beta^{2}}\sum_{j=2}^{J}\Lambda_{j}\partial_{j}\Omega$
$\displaystyle-\frac{2}{\beta}\sum_{j=2}^{J}\Lambda_{j}\partial_{1}\partial_{j}\Omega-\frac{1}{\beta}\partial_{1}^{2}\Omega$
$\displaystyle-\frac{1}{\beta}\sum_{i=2}^{J}\sum_{j=2}^{J}\Lambda_{i}\Lambda_{j}\partial_{i}\partial_{j}\Omega$
$\displaystyle\partial_{\beta}\partial_{\Lambda_{i}}\Delta$
$\displaystyle=-\partial_{1}\partial_{i}\Omega-\sum_{j=2}^{J}\Lambda_{j}\partial_{i}\partial_{j}\Omega$
$\displaystyle\partial_{\Lambda_{i}}\partial_{\Lambda_{j}}\Delta$
$\displaystyle=-\beta\partial_{i}\partial_{j}\Omega$ (62)

To demonstrate positive definiteness, it is necessary to show

$\displaystyle\forall
a\in\mathbb{R}^{J}:\sum_{k=1}^{J}\sum_{l=1}^{J}a_{k}a_{l}\partial_{k}\partial_{l}\Delta\geq
0$ (63)

We need to prove that the following expression is for all $a$ positive

$\displaystyle
a_{1}^{2}\partial^{2}_{\beta}\Delta+2\sum_{k=2}^{J}a_{1}a_{k}\partial_{\beta}\partial_{\Lambda_{k}}\Delta+\sum_{k=2}^{J}\sum_{l=2}^{J}a_{k}a_{l}\partial_{\Lambda_{k}}\partial_{\Lambda_{l}}\Delta$
(64)

By sorting the terms in eq. (64), we obtain the following two contributions
expressed in relation to $\Omega$

$\displaystyle\frac{2a_{1}^{2}}{k\beta^{3}}\left(-k\Omega+k\beta
E+\beta\sum_{j=2}^{J}\Lambda_{j}r_{j}\right)$ (65)

and

$\displaystyle-\frac{1}{\beta}\bigg{(}$ $\displaystyle
a_{1}^{2}\partial_{1}^{2}\Omega+2\sum_{j=2}^{J}a_{1}(a_{1}\Lambda_{j}+a_{j}\beta)\partial_{1}\partial_{j}\Omega$
$\displaystyle+\sum_{i=2}^{J}\sum_{j=2}^{J}(a_{1}\Lambda_{i}+a_{i}\beta)(a_{1}\Lambda_{j}+a_{j}\beta)\partial_{i}\partial_{j}\Omega\bigg{)}$
(66)

The contribution of the first term eq. (65) is proportional to $S$

$\displaystyle\frac{2a_{1}^{2}}{k\beta^{3}}S(E,\Lambda)$ (67)

The second term eq. (66) is related to the negative definiteness of the
Hessian of $\Omega$ in eq. (20a), with the specific choice of
$b\coloneqq(a_{1},\\{a_{1}\Lambda_{j}+a_{j}\beta\\}_{j=2}^{J})\in\mathbb{R}^{J}$:

$\displaystyle-\frac{1}{\beta}\underbrace{\sum_{k=1}^{J}\sum_{l=1}^{J}b_{k}b_{l}\partial_{k}\partial_{l}\Omega}_{\leq
0}$ (68)

For $\beta\geq 0$, the two contributions eqs. (67) and (68) are positive.
Therefore, $\Delta$ is indeed convex.

## References

  * Callen [1991] H. Callen,  _Thermodynamics and an Introduction to Thermostatistics_ , 2nd ed. (Wiley, 1991) p. 512.
  * Neumaier and Westra [2011] A. Neumaier and D. Westra, Classical and quantum mechanics via lie algebras (2011), arXiv:0810.1019 [quant-ph] .
  * LANDAU and LIFSHITZ [1980a] L. LANDAU and E. LIFSHITZ, Chapter i - the fundamental principles of statistical physics, in  _Statistical Physics (Third Edition)_, Course of Theoretical Physics, Vol. 5, edited by L. LANDAU and E. LIFSHITZ (Pergamon, 1980) third edition ed., pp. 1–33.
  * Macke [1963] W. Macke,  _Ein Lehrbuch der theoretischen Physik : Thermodynamik und Statistik / Wilhelm Macke_ , 2nd ed. (Geest, Portig, 1963).
  * Schuller [2012] F. P. Schuller, Theoretische statistische mechanik (2012), lecture notes.
  * Fultz [2020] B. Fultz,  _Phase Transitions in Materials_ (Cambridge University Press, 2020).
  * Khomskii [2010] D. I. Khomskii,  _BASIC ASPECTS OF THE QUANTUM THEORY OF SOLIDS_ (Cambridge University Press, 2010).
  * LANDAU and LIFSHITZ [1980b] L. LANDAU and E. LIFSHITZ, Chapter iii - the gibbs distribution, in  _Statistical Physics (Third Edition)_, Course of Theoretical Physics, Vol. 5, edited by L. LANDAU and E. LIFSHITZ (Pergamon, 1980) third edition ed., pp. 79–110.
  * Schuller [2015] F. P. Schuller, Lectures on quantum theory (elite graduate programme) 2015 -measurements (2015), lecture notes.
  * Tamarkin and von Neumann [1935] J. D. Tamarkin and J. von Neumann, Mathematische Grundlagen der Quantenmechanik., The American Mathematical Monthly 42, 237 (1935).


Center for coherence and quantum optics, Department of Physics, University of
Rochester IR,NMR,UV

# Interplay of trapped species and absence of electron capture in Moiré
heterobilayers

Arnab Barman Ray  The Institute of Optics, University of Rochester  Arunabh
Mukherjee  The Institute of Optics, University of Rochester  Liangyu Qiu  The
Institute of Optics, University of Rochester  Renee Sailus  Arizona State
University  Sefaattin Tongay  Arizona State University  Anthony Nickolas
Vamivakas  The Institute of Optics, University of Rochester
<EMAIL_ADDRESS>The Institute of Optics, University of Rochester

###### Abstract

Moiré heterobilayers host interlayer excitons in a natural, periodic array of
trapping potentials. Recent work has elucidated the structure of the trapped
interlayer excitons and the nature of photoluminescence (PL) from trapped and
itinerant charged complexes such as interlayer trions in these structures. In
this paper, our results serve to add to the understanding of the nature of PL
emission and explain its characteristic blueshift with increasing carrier
density, along with demonstrating a significant difference between the
interlayer exciton-trion conversion efficiency as compared to both localized
and itinerant intra-layer species in conventional monolayers. Our results show
the absence of optical generation of trions in these materials, which we
suggest arises from the highly localized, near sub-nm confinement of trapped
species in these Moiré potentials.

###### keywords:

American Chemical Society, LaTeX

##  1 Introduction

Heterostructures, made by stacking monolayers of transition metal
dichalcogenides1, 2, 3, 4, 5, 6, 7 (TMDCs), as well as other materials, such
as insulating hexagonal boron nitride(hBN)8, 9, have attracted significant
scientific interest over the last decade. The possibilities of constructing
different kinds of condensed matter systems10, 11, 12 through the layer degree
of freedom and the choice of material have been a cornerstone of research in
this direction. In recent years, it was found that bilayers, consisting of two
monolayers of either the same (homobilayer) or different (heterobilayer)
monolayers can host luminescent long-lived interlayer excitons13, 14. The
longer lifetimes of interlayer excitons in these heterostructures, as opposed
to intralayer excitons in monolayers, have made them a material of choice for
excitonic applications15, 16. More recently, it has been found that two
monolayers in contact with each other in a heterobilayer (hBL), develop a
periodically varying potential that can trap these interlayer excitons17, 18,
19, 20. The structures, called Moiré superlattices, have a period that depends
critically on the twist angle between these layers and the mismatch between
the lattice period of the two layers themselves21, 22. These hBLs were shown
to host single-photon quantum emitters at the Moiré trapping sites, which also
offer a degree of “programmability” through the use of magnetic and electric
fields, which can be used to tune the emission wavelengths23 and
polarization24.

$\text{MoSe}_{2}/\text{WSe}_{2}$ hBLs is one of the most well studied TMDC
Moiré systems25, 26. They are characterized by a large Moiré period ( 100 nm)
with near $0^{\circ}$ alignment, owing to their small lattice mismatch between
the two monolayer. More recently, isolated trapped species have been observed,
trions, bi-excitons, and evidence for higher order charged complexes have
emerged27, 28, 29. Beyond isolated emitters, at higher excitation intensities,
different species of trions have been identified in this system30. Moiré
trions are very interesting as they allow a higher degree of nonlinearity in
their interactions with light through a phase-space filling effect combined
with Moiré localization effects31. They may have applications for nonlinear
photonic devices in quantum computing applications32, 33. In this regard, a
thorough understanding of the physics of Moiré trions is desirable.

The photoluminescence (PL) emission from interlayer excitons in
$\text{MoSe}_{2}/\text{WSe}_{2}$ Moiré heterobilayers has been studied in some
detail, and three broad phases of excited state species were reported from
diffusion-based studies - trapped excitons, itinerant excitons, and electron-
hole plasmas at very high excitation intensities equivalent to carrier
densities of the order of $10^{13}\,\text{cm}^{-2}$ 4, 34. However, at
intermediate carrier densities in the range of
$10^{10}-10^{11}\,\text{cm}^{-2}$, when the spectral lines from individual
trapped emitters are no longer isolatable due to power broadening and higher
densities of trapped emitters, the dynamics between different species are not
clear. This state of the system, which we call the quantum ensemble regime30,
represents emission from a collection of different trapped species across the
excitation spot.

Our results investigate this ensemble, and we observe, by normalizing the PL,
that trions, excitons, and bi-excitons exist in this regime and are seen as
spectrally separated emission bands. We find a spectral weight transfer across
the different species with increasing optical excitation density. The strong
correlation of the trion PL with the amount of electrostatically doped
electrons in the system indicates an absence of trion-exciton conversion at
higher excitation fluxes typical of free and trapped intralayer excitons in
monolayers35, 36, 37. We note that the absence of optically generated trions
is a feature that distinguishes Moiré systems from conventional monolayers
that have been studied so far, with itinerant38, 35 or defect-localized intra-
layer excitons36. The blue shift observed in the PL at lower excitation
intensities has been attributed to the dipolar repulsion between interlayer
excitons across Moiré trapping sites30. However, our results show that most of
the blue-shift that has been observed consistently so far arise primarily from
a spectral weight transfer from one species to the other with increasing
excitation intensity.

##  2 Sample and Methods

We use a dry transfer technique to assemble a double hBN encapsulated
$\text{MoSe}_{2}-\text{WSe}_{2}$ single-gated heterostructure with a PC stamp
under a microscope. The hBN was obtained from 2D semiconductors. The bulk
$\text{MoSe}_{2}$ was n-type from HQ Graphene, and the WSe2 was prepared at
ASU. The monolayers were aligned along precise straight edges, which allowed
us to make luminescent hBLs, but without any control on whether the resulting
stack would be R-type or H-type21, although magnetic measurements in this
sample confirm from the observed g-factors, that the sample is R-type39. We
use an n-doped monolayer to observe trions without any electrostatic doping.
We assemble the hBL on top of a chip with 285 nm of thermally grown
$\text{SiO}_{2}$ on Si. The back gate consists of an FLG in contact with one
of the 50 nm gold electrodes. The monolayers are also in contact with another
electrode, allowing us to bias the device as a parallel plate capacitor. An
accurate voltage source (ANC 300 controller) is used to bias the device.
Figure 1(b) shows an optical micrograph of the device under consideration.

The measurements were carried out using a home-built confocal microscope. A
PID stabilized 532 nm DPSS laser is focused into a sub-um diameter spot using
a $0.82$ NA objective lens in a closed-cycle cryostat (AttoDry 1000) at 4K
equipped with a super-conducting magnet. The PL emitted is collected by the
same objective and coupled into a multimode fiber. The collected PL is
analyzed using a Princeton Instruments spectrometer (Acton SP-2750i) and an
LN2 cooled Pylon CCD camera. We used an intensity calibrator (IntelliCal,
Princeton Instruments) to account for the drop in quantum efficiency of the Si
CCD camera at near-IR wavelengths.

Figure 1: (a)A schematic of the kind of devices investigated, (b) optical
image of the first sample, the scale bar is 10$\mu$m, (c) PL from interlayer
excitons and trions showing a transition with increased doping voltage at an
excitation intensity of 20 nW, the insets are a plot of the PL spectra at
$V_{G}=0.8\,\text{V}$(top) $V_{G}=0.0\,\text{V}$(bottom), (d) evolution of the
PL with increasing laser intensity, with insets at $1$ $\mu$W (left) and $9$
$\mu$W (right).

##  3 Results

First, we investigate the PL from the sample at low carrier densities, when
the narrow emission from the individual emitters is well differentiated. The
PL we investigate from this sample (R-type) arises from the singlet exciton40,
41. Figure 1(c) shows a voltage sweep, where the voltage is systematically
incremented until we see a sharp transition from trapped interlayer excitons
to trions. The narrow emission peaks are characteristics of these quantum
emitters with individual linewidths $<\,100\mu\text{eV}$. We then focus on how
the PL evolves with increasing laser intensity in Fig. 1(d). We note a blue-
shift in the observed PL by about 8 meV. The linewidth of the emission is
around 5-6 meV. We note the non-linear evolution of the PL with excitation
intensity. For the highest excitation intensities considered in this paper, we
estimate a charge carrier density of $~{}5\times 10^{11}\,\text{cm}^{-2}$,
well within the regime where the emission is dominated by moiré trapped
excitonic species34. We arrive at this estimate by using the steady state
equation, $n=\alpha I\tau$ and a carrier lifetime of
$\tau\,=\,2\,\text{ns}$13, an absorption coefficient42 of $\alpha=0.10$ at
$2.33\,\text{eV}$ in a spot size of area $1\,\mu m^{2}$.

We normalize the PL spectra in Fig. 2(a) to gain further insight. We normalize
the PL emission at a given excitation intensity by dividing the spectrum by
the observed peak counts. We see that normalizing the PL resolves the emission
into three distinct bands. The energy difference between the first two bands
is obtained by subtracting the energy difference between the center of the two
bands and is found to be around $5-6\,\text{meV}$. The energy difference
between the second and third band is around $3-4\,\text{meV}$. We note that
these values are consistent with reported values of the binding energy for
trion29, 30, and the additional energy that the trapped bi-exciton exhibits
due to intra-trap dipolar repulsion respectively27. The binding energies for
the trion are much lower than are seen for intra-layer trions due to the
effects of localization on the trion wavefunction. Similarly, the spatial
proximity of two excitons in the same Moiré trap and the extreme localization
causes the bi-exciton to emit at a higher energy.

Figure 2:  Normalized PL spectra vs power on sample at doping voltages of (a)
$V_{G}=0.6\,V$, and (b) $V_{G}=0.8\,V$,(c) traces of the peak PL intensity
with excitation power at different doping voltages, (d) power law fits of the
integrated PL intensities of the three species, $I_{1}$, $I_{2}$ and $I_{3}$
correspond to trions, excitons, and bi-excitons respectively.

To confirm the nature of these bands, we fit the bands with separate
lorentzians with linewidths of $2\,\text{meV}$ each. The integrated PL
intensities obtained from these fits are than plotted as a function of laser
power and subjected to a linear fit in Fig. 2(d). The highest energy band
exhibits a growth coefficient of $\gamma_{3}\,>\,1$. This superlinear growth
confirms the bi-excitonic nature of the highest energy band. On the other
hand, the doping voltage dependence of the first band confirms that it has a
trion origin. The polarity of the applied doping voltages confirms the
negatively charged nature of the trions. We note that the small linewidths of
each emission band $\sim\,2\,\text{meV}$ is comparable to the smallest
linewidths reported so far, reflecting on the quality of the materials used
and the sample quality. Our results show that at higher excitation densities,
the PL emission from these samples is predominantly bi-excitonic in origin.
Moreover, the differences between the energies of the excitons and bi-
excitons, which arise from intra-trap dipolar interactions, can explain most
of the blue shift that is seen in the PL from these samples.

We next shift our attention to how the PL evolves with power as a function of
the doping density. We trace the peak PL energies at different gate voltages
in Fig. 2(c). We are able to see the shifts in spectral weight from trions to
excitons and bi-excitons with increasing laser power. We also notice a DC
stark effect, due to the non-zero electric field across the sample. The most
exciting feature we observed was that the trion-to-exciton spectral weight
jump was a function of the doping density of the sample, and the trion
emission dominated the PL spectrum up to higher carrier densities in the
sample for larger doping densities.

In monolayers with itinerant or defect-localized excitons, at higher laser
fluences, the exciton-trion ratio decreases38, 35, 36. This is due to the
prevalence of unbound free carriers. Trions can then form from excitons
through electron capture37 from a sea of unbound charge carriers. This effect
tends to increase with increased density of unbound free carriers at higher
laser fluences, as the exciton-electron interaction increases43. Localization,
as seen in the case of excitonic quantum emitters based on defects36, 43,
still allows this exciton-electron interaction. However, our studies reveal
the absence of such a mechanism for Moiré localized excitons, suggesting a
more robust localization, with the reduced spatial extent and center-of-mass
motion44 resulting in a reduced interaction with the sea of free carriers.

To understand the interplay between electrostatic doping and laser-created
electron-hole carriers, we investigate how the exciton-trion transition for
the case of a doping voltage sweep shifts with increasing excitation
intensity. In Fig. 3 (a) and (b), we present the PL spectrum in a voltage
sweep for two different excitation intensities. We see that the exciton-trion
crossover takes place at a larger doping density at higher laser intensities.
We plot the peak PL energy as a function of gate voltage for different
excitation intensities in Fig. 3 (c); this allows us to see that the
transition voltage shifts with increasing laser power. Figure 3 (d) plots the
transition voltages as a function of laser power. We see that at higher laser
fluences, it takes a larger amount of dopant electrons to shift the dominant
optical transition of the system to trions. The linear correlation between the
doping voltage and the transition voltage suggests that any trions formed in
the system are through the presence of doped electrons sitting at the Moiré
traps45.

Figure 3:  PL spectra with vs doping voltage at excitation pwoers of (a) 25
nW, and (b) 3.62 $\mu$W (c) traces of the peak PL intensity with doping
voltage at different excitation powers, (d) transition voltages as a function
of power on sample, (e) a cartoon illustrating the difference between highly
localized excitons in a Moire trap and an itinerant exciton in a monolayer,
yellow spheres are free carriers and the blue hat profiles are representatives
of the respective excitonic wavefunctions.

To summarize, our results paint the following picture. We start with an array
of Moiré traps with a constant density of doped electrons at the trap sites.
As we excite the system with a laser, the trion states- the lowest excited
states, start filling up. This is when the PL is mostly trion-ic in origin. At
higher laser fluences, as all the doped electrons are bound as trions, the
absence of electron capture makes it impossible for the population of trions
to grow; hence the Moire traps start filling up with excitons. When all the
traps are nearly filled, bi-excitonic emission takes over. We note that the
presence of charged bi-excitonic species cannot be ruled out; however, further
studies are required to mount evidence for their existence.

##  4 Conclusions and Future Work

Our work illustrates the interplay of different species in the PL emission
from $\text{MoSe}_{2}/\text{WSe}_{2}$ Moiré heterobilayers. Normalizing the PL
spectra and tracing its evolution with excitation power shows that most of the
observed blueshift in these samples can be attributed to the spectral weight
transfer in the PL across different species and hence to intra-trap dipole-
dipole interactions as opposed to interactions across different Moire sites.
We find that at higher laser fluences (carrier densities
$<\,10^{12}\,\text{cm}^{-2}$), the PL is dominated by bi-excitonic emission.
By analyzing the trion-exciton spectral weight, we also show that Moiré
localization leads to an absence of electron capture in these systems, which
hints at a qualitative difference in the strength of localization over
conventional defect-based quantum emitters in monolayers. Investigating the
time dynamics of trion formation and a rigorous rate equation analysis of
these systems may shed more light on the exciton-electron interactions. Two-
dimensional Fourier spectroscopy may also reveal how different trapped species
are coupled to each other by analyzing the cross-couplings and may shed more
light on the excited state landscape in these systems.

##  5 Acknowledgments

This work was supported by AFOSR FA9550-19-1-0074 from the Cornell Center for
Materials Research with funding from the NSF MRSEC program (DMR-1719875). S.T
acknowledges NSF CMMI 2129412 and NSF DMR 2111812.

## References

  * He et al. 2014 He, K.; Kumar, N.; Zhao, L.; Wang, Z.; Mak, K. F.; Zhao, H.; Shan, J. Tightly bound excitons in MonolayerWSe2. _Phys. Rev. Lett._ 2014, _113_
  * Mak et al. 2013 Mak, K. F.; He, K.; Lee, C.; Lee, G. H.; Hone, J.; Heinz, T. F.; Shan, J. Tightly bound trions in monolayer MoS2. _Nat. Mater._ 2013, _12_ , 207–211
  * Jones et al. 2013 Jones, A. M.; Yu, H.; Ghimire, N. J.; Wu, S.; Aivazian, G.; Ross, J. S.; Zhao, B.; Yan, J.; Mandrus, D. G.; Xiao, D.; Yao, W.; Xu, X. Optical generation of excitonic valley coherence in monolayer WSe2. _Nat. Nanotechnol._ 2013, _8_ , 634–638
  * Barman Ray et al. 2022 Barman Ray, A.; Liang, K.; Vamivakas, A. N. Valley engineering electron-hole liquids in transition metal dichalcogenide monolayers. _Phys. Rev. B_ 2022, _106_ , 045206
  * Splendiani et al. 2010 Splendiani, A.; Sun, L.; Zhang, Y.; Li, T.; Kim, J.; Chim, C.-Y.; Galli, G.; Wang, F. Emerging Photoluminescence in Monolayer MoS2. _Nano Letters_ 2010, _10_ , 1271–1275
  * Castellanos-Gomez 2016 Castellanos-Gomez, A. Why all the fuss about 2D semiconductors? _Nature Photonics_ 2016, _10_ , 202–204
  * Mak and Shan 2016 Mak, K. F.; Shan, J. Photonics and optoelectronics of 2D semiconductor transition metal dichalcogenides. _Nature Photonics_ 2016, _10_ , 216–226
  * Cadiz et al. 2017 Cadiz, F. et al. Excitonic Linewidth Approaching the Homogeneous Limit in MoS2 -Based van der Waals Heterostructures. _Phys. Rev. X._ 2017, _7_
  * Taniguchi and Watanabe 2007 Taniguchi, T.; Watanabe, K. Synthesis of high-purity boron nitride single crystals under high pressure by using Ba–BN solvent. _Journal of Crystal Growth_ 2007, _303_ , 525–529
  * Gu et al. 2022 Gu, J.; Ma, L.; Liu, S.; Watanabe, K.; Taniguchi, T.; Hone, J. C.; Shan, J.; Mak, K. F. Dipolar excitonic insulator in a moiré lattice. _Nature Physics_ 2022, _18_ , 395–400
  * Chen et al. 2022 Chen, D. et al. Excitonic insulator in a heterojunction moiré superlattice. _Nature Physics_ 2022, _18_ , 1171–1176
  * Tang et al. 2020 Tang, Y.; Li, L.; Li, T.; Xu, Y.; Liu, S.; Barmak, K.; Watanabe, K.; Taniguchi, T.; MacDonald, A. H.; Shan, J.; Mak, K. F. Simulation of Hubbard model physics in WSe2/WS2 moiré superlattices. _Nature_ 2020, _579_ , 353–358
  * Rivera et al. 2015 Rivera, P.; Schaibley, J. R.; Jones, A. M.; Ross, J. S.; Wu, S.; Aivazian, G.; Klement, P.; Seyler, K.; Clark, G.; Ghimire, N. J.; Yan, J.; Mandrus, D. G.; Yao, W.; Xu, X. Observation of long-lived interlayer excitons in monolayer MoSe2–WSe2 heterostructures. _Nat. Commun._ 2015, _6_
  * Miller et al. 2017 Miller, B.; Steinhoff, A.; Pano, B.; Klein, J.; Jahnke, F.; Holleitner, A.; Wurstbauer, U. Long-lived direct and indirect interlayer excitons in van der Waals heterostructures. _Nano Lett._ 2017, _17_ , 5229–5237
  * Unuchek et al. 2018 Unuchek, D.; Ciarrocchi, A.; Avsar, A.; Watanabe, K.; Taniguchi, T.; Kis, A. Room-temperature electrical control of exciton flux in a van der Waals heterostructure. _Nature_ 2018, _560_ , 340–344
  * Sun et al. 2022 Sun, Z.; Ciarrocchi, A.; Tagarelli, F.; Gonzalez Marin, J. F.; Watanabe, K.; Taniguchi, T.; Kis, A. Excitonic transport driven by repulsive dipolar interaction in a van der Waals heterostructure. _Nat. Photonics_ 2022, _16_ , 79–85
  * Jin et al. 2019 Jin, C.; Regan, E. C.; Yan, A.; Iqbal Bakti Utama, M.; Wang, D.; Zhao, S.; Qin, Y.; Yang, S.; Zheng, Z.; Shi, S.; Watanabe, K.; Taniguchi, T.; Tongay, S.; Zettl, A.; Wang, F. Observation of moiré excitons in WSe2/WS2 heterostructure superlattices. _Nature_ 2019, _567_ , 76–80
  * Förg et al. 2021 Förg, M.; Baimuratov, A. S.; Kruchinin, S. Y.; Vovk, I. A.; Scherzer, J.; Förste, J.; Funk, V.; Watanabe, K.; Taniguchi, T.; Högele, A. Moiré excitons in MoSe2-WSe2 heterobilayers and heterotrilayers. _Nat. Commun._ 2021, _12_
  * Yu et al. 2017 Yu, H.; Liu, G.-B.; Tang, J.; Xu, X.; Yao, W. Moiré excitons: From programmable quantum emitter arrays to spin-orbit–coupled artificial lattices. _Sci. Adv._ 2017, _3_
  * Seyler et al. 2019 Seyler, K. L.; Rivera, P.; Yu, H.; Wilson, N. P.; Ray, E. L.; Mandrus, D. G.; Yan, J.; Yao, W.; Xu, X. Signatures of moiré-trapped valley excitons in MoSe2/WSe2 heterobilayers. _Nature_ 2019, _567_ , 66–70
  * Mak and Shan 2022 Mak, K. F.; Shan, J. Semiconductor moiré materials. _Nat. Nanotechnol._ 2022, _17_ , 686–695
  * Liu et al. 2021 Liu, Y.; Zeng, C.; Yu, J.; Zhong, J.; Li, B.; Zhang, Z.; Liu, Z.; Wang, Z. M.; Pan, A.; Duan, X. Moiré superlattices and related moiré excitons in twisted van der Waals heterostructures. _Chem. Soc. Rev._ 2021, _50_ , 6401–6422
  * Baek et al. 2020 Baek, H.; Brotons-Gisbert, M.; Koong, Z. X.; Campbell, A.; Rambach, M.; Watanabe, K.; Taniguchi, T.; Gerardot, B. D. Highly energy-tunable quantum light from moiré-trapped excitons. _Sci. Adv._ 2020, _6_
  * Ciarrocchi et al. 2019 Ciarrocchi, A.; Unuchek, D.; Avsar, A.; Watanabe, K.; Taniguchi, T.; Kis, A. Polarization switching and electrical control of interlayer excitons in two-dimensional van der Waals heterostructures. _Nat. Photonics_ 2019, _13_ , 131–136
  * Brem et al. 2020 Brem, S.; Linderälv, C.; Erhart, P.; Malic, E. Tunable Phases of Moiré Excitons in van der Waals Heterostructures. _Nano Letters_ 2020, _20_ , 8534–8540
  * Sigl et al. 2022 Sigl, L.; Troue, M.; Katzer, M.; Selig, M.; Sigger, F.; Kiemle, J.; Brotons-Gisbert, M.; Watanabe, K.; Taniguchi, T.; Gerardot, B. D.; Knorr, A.; Wurstbauer, U.; Holleitner, A. W. Optical dipole orientation of interlayer excitons in ${\mathrm{MoSe}}_{2}-{\mathrm{WSe}}_{2}$ heterostacks. _Phys. Rev. B_ 2022, _105_ , 035417
  * Li et al. 2020 Li, W.; Lu, X.; Dubey, S.; Devenica, L.; Srivastava, A. Dipolar interactions between localized interlayer excitons in van der Waals heterostructures. _Nat. Mater._ 2020, _19_ , 624–629
  * Liu et al. 2021 Liu, E.; Barré, E.; van Baren, J.; Wilson, M.; Taniguchi, T.; Watanabe, K.; Cui, Y.-T.; Gabor, N. M.; Heinz, T. F.; Chang, Y.-C.; Lui, C. H. Signatures of moiré trions in WSe2/MoSe2 heterobilayers. _Nature_ 2021, _594_ , 46–50
  * Wang et al. 2021 Wang, X.; Zhu, J.; Seyler, K. L.; Rivera, P.; Zheng, H.; Wang, Y.; He, M.; Taniguchi, T.; Watanabe, K.; Yan, J.; Mandrus, D. G.; Gamelin, D. R.; Yao, W.; Xu, X. Moiré trions in MoSe2/WSe2 heterobilayers. _Nature Nanotechnology_ 2021, _16_ , 1208–1213
  * Brotons-Gisbert et al. 2021 Brotons-Gisbert, M.; Baek, H.; Campbell, A.; Watanabe, K.; Taniguchi, T.; Gerardot, B. D. Moiré-Trapped Interlayer Trions in a Charge-Tunable ${\mathrm{WSe}}_{2}/{\mathrm{MoSe}}_{2}$ Heterobilayer. _Phys. Rev. X_ 2021, _11_ , 031033
  * Zhang et al. 2021 Zhang, L.; Wu, F.; Hou, S.; Zhang, Z.; Chou, Y.-H.; Watanabe, K.; Taniguchi, T.; Forrest, S. R.; Deng, H. Van der Waals heterostructure polaritons with moiré-induced nonlinearity. _Nature_ 2021, _591_ , 61–65
  * Emmanuele et al. 2020 Emmanuele, R. P. A.; Sich, M.; Kyriienko, O.; Shahnazaryan, V.; Withers, F.; Catanzaro, A.; Walker, P. M.; Benimetskiy, F. A.; Skolnick, M. S.; Tartakovskii, A. I.; Shelykh, I. A.; Krizhanovskii, D. N. Highly nonlinear trion-polaritons in a monolayer semiconductor. _Nature Communications_ 2020, _11_ , 3589
  * Kyriienko et al. 2020 Kyriienko, O.; Krizhanovskii, D. N.; Shelykh, I. A. Nonlinear Quantum Optics with Trion Polaritons in 2D Monolayers: Conventional and Unconventional Photon Blockade. _Phys. Rev. Lett._ 2020, _125_ , 197402
  * Wang et al. 2021 Wang, J.; Shi, Q.; Shih, E.-M.; Zhou, L.; Wu, W.; Bai, Y.; Rhodes, D.; Barmak, K.; Hone, J.; Dean, C. R.; Zhu, X.-Y. Diffusivity Reveals Three Distinct Phases of Interlayer Excitons in ${\mathrm{MoSe}}_{2}/{\mathrm{WSe}}_{2}$ Heterobilayers. _Phys. Rev. Lett._ 2021, _126_ , 106804
  * Kesarwani et al. 2022 Kesarwani, R.; Simbulan, K. B.; Huang, T.-D.; Chiang, Y.-F.; Yeh, N.-C.; Lan, Y.-W.; Lu, T.-H. Control of trion-to-exciton conversion in monolayer WS<sub>2</sub> by orbital angular momentum of light. _Science Advances_ 2022, _8_ , eabm0100
  * Chakraborty et al. 2018 Chakraborty, C.; Qiu, L.; Konthasinghe, K.; Mukherjee, A.; Dhara, S.; Vamivakas, N. 3D Localized Trions in Monolayer WSe2 in a Charge Tunable van der Waals Heterostructure. _Nano Letters_ 2018, _18_ , 2859–2863
  * Portella-Oberli et al. 2009 Portella-Oberli, M. T.; Berney, J.; Kappei, L.; Morier-Genoud, F.; Szczytko, J.; Deveaud-Plédran, B. Dynamics of Trion Formation in ${\mathrm{In}}_{x}{\mathrm{Ga}}_{1-x}\mathrm{As}$ Quantum Wells. _Phys. Rev. Lett._ 2009, _102_ , 096402
  * Gao et al. 2016 Gao, F.; Gong, Y.; Titze, M.; Almeida, R.; Ajayan, P. M.; Li, H. Valley trion dynamics in monolayer ${\mathrm{MoSe}}_{2}$. _Phys. Rev. B_ 2016, _94_ , 245413
  * 39 Supplementary Information.
  * Yu et al. 2018 Yu, H.; Liu, G.-B.; Yao, W. Brightened spin-triplet interlayer excitons and optical selection rules in van der Waals heterobilayers. _2D Materials_ 2018, _5_ , 035021
  * Joe et al. 2021 Joe, A. Y. et al. Electrically controlled emission from singlet and triplet exciton species in atomically thin light-emitting diodes. _Phys. Rev. B_ 2021, _103_ , L161411
  * Rigosi et al. 2015 Rigosi, A. F.; Hill, H. M.; Li, Y.; Chernikov, A.; Heinz, T. F. Probing Interlayer Interactions in Transition Metal Dichalcogenide Heterostructures by Optical Spectroscopy: MoS2/WS2 and MoSe2/WSe2. _Nano Letters_ 2015, _15_ , 5033–5038
  * Singh et al. 2016 Singh, A. et al. Trion formation dynamics in monolayer transition metal dichalcogenides. _Phys. Rev. B_ 2016, _93_ , 041401
  * Karni et al. 2022 Karni, O. et al. Structure of the moiré exciton captured by imaging its electron and hole. _Nature_ 2022, _603_ , 247–252
  * Baek et al. 2021 Baek, H.; Brotons-Gisbert, M.; Campbell, A.; Vitale, V.; Lischner, J.; Watanabe, K.; Taniguchi, T.; Gerardot, B. D. Optical read-out of Coulomb staircases in a moiré superlattice via trapped interlayer trions. _Nature Nanotechnology_ 2021, _16_ , 1237–1243