# Thermodynamic equilibrium and detailed balance

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## Toward a description of NESS?

In this section, we review very brie y several tools that have been developed to describe out-of-equilibrium systems.

Generalized detailed balance and uctuation theorem

Even if the detailed balance does not hold for a general Markovian process, it is possible to formulate a generalization of it that is always satis ed. For the sake of simplicity, we will consider the particular case where the system is in contact with two reservoirs at di erent inverse temperatures 1 and 2 and can exchange energy with these reservoirs. The discussion below can be adapted to the case of more than two reservoirs and of di erent physical quantities (such as charges, momentum, particles) exchanged with the reservoirs. The generalized detailed balance relation reads 1 E1 2 E2 ; (I.C.1) where E1 (respectively E2) stands for the energy exchanged with reservoir 1 (respectively during the transition from con guration C to con guration C0. We observe that formula (I.C.1) is a generalization of the detailed balance relation (I.B.32), which is easily recovered when 1 = 2. Relation (I.C.1) can be physically justi ed by considering the system and the two reservoirs as an isolated system, whose dynamics should reach the microcanonical distribution. Writing the detailed balance relation for this whole system, and assuming that the energy exchanged is small in comparison to the total energy of the reservoirs, yields the generalized detailed balance condition (see for instance [16, 17] for details).
This generalized detailed balance relation can be interpreted as the root of the uctuation theorem, see subsection 3.

Dynamic observables and deformed Markov matrix

In a system at thermodynamic equilibrium, we saw that the stationary distribution is given by the Boltzmann distribution S(C) e E(C) (I.C.2) for a system in contact with a reservoir at inverse temperature . It allows us to de ne the free energy, which is simply the cumulant generating function of the energy observable E. This free energy is a very e cient tool to describe the properties of the system. We would like to build the same kind of tool to describe out-of-equilibrium systems and more particularly non-equilibrium stationary states. By analogy with equilibrium systems, the generalized balance condition (I.C.1) suggests to study the uctuations of the energy exchanged with the reservoirs and to construct the generating function of this observable.
This can be formalized as follows. We are interested in studying an observable O, whose value O(C ! C0) depends on the transition C ! C0 under consideration. It can be for instance the energy exchanged with a particular reservoir during the transition or the number of particles injected at a particular place during the transition. We would like to compute the uctuations of the observable O in the stationary state. To give a precise mathematical meaning to this idea, we rst need to give a sense to the value of the observable in the stationary state. One way to achieve that is to begin by de ning the value of the observable on a whole path history. For the path history fC(t)g0 t T de ned in subsection a), which explores successively the con gurations C1; : : : ; Cn (with transitions occurring at times t1; : : : ; tn 1), we de ne OT (fC(t)g) = O(C1 ! C2) + + O(Cn 1 ! Cn): (I.C.3)
The value OT depends on the time evolution, or path history, of the system (which is governed by the master equation (I.B.15)) on the interval [0; T ]. It is a random variable. We denote by PT (O) the probability that OT = O. Our goal is to study this probability distribution in the large time T limit. To determine the time evolution of this probability distribution, we have to consider the more precise quantity PT (O; C), which denotes the joint probability for the system to be in con guration C at time T and for the observable OT to be equal to O. The value of PT (O) will be simply recovered through the relation PT (O) = C PT (O; C).
This cumulant generating function in the stationary state turns out to be an e cient tool to characterize the macroscopic behavior of a physical system in the large size limit (thermodynamic limit, see chapter V), if the observable O is correctly chosen. In particular the singularities of this function (or of the associated large deviation function, see subsection 4) could determine the phase transitions of the model [18{20].
In this manuscript, we will be interested in interacting particles systems evolving on a one dimensional lattice. A physical quantity of prime interest in such models is the (algebraic) number of particles which cross a particular bond on the lattice during a transition. This observable is indeed representative of the non-equilibrium aspects of these interacting particles systems. It provides a quantitative way to characterize the particles current and its uctuation in the stationary state, through the computation of the cumulant generating function. In chapter III and chapter IV we will address the problem of computing exactly this generating function for particular models.
This formalism allows also to state an elegant symmetry on the uctuations of the entropy production, which we present in the subsection below.
This is known as the Gallavotti-Cohen symmetry [21{23]. We proved here the symmetry involving the entropy production (which holds for any Markov chain) but similar relations can also be derived for other physical observables, depending on the model under consideration. For instance, we will encounter such a symmetry in chapter IV while studying the cumulant generating function of the particle current for the asymmetric simple exclusion process.
The symmetry on the generating function translates into a symmetry on the large deviation function of the entropy production (see subsection 4 for de nition and details), which can be de ned as G(s) = inf[ s E( )]: (I.C.18)
This uctuation theorem (and its generalization to other physical observables, depending on the model under consideration) has very useful implications. It yields near equilibrium the Einstein uctuation-dissipation relations, the Onsager reciprocity relations and the Kubo linear response theory, see for instance .

### Large deviation functions

In the last decades, the large deviation theory has proven to be a very e cient framework to deal with equilibrium but also non-equilibrium systems. We present brie y below the main tools (from a statistical physicist point of view) associated to this theory. The reader is invited to read the very useful review  for details.

Large deviation principle

The large deviation theory can be intuitively introduced as a framework to evaluate the proba-bilities of rare events. From a mathematical point of view, it could be thought as a re nement of the law of large numbers. From a physical point of view, the large deviation functions are seen as possible generalizations of thermodynamic potentials in the out-of-equilibrium context.
De nition C.5. We consider (Sn) a sequence of random variables taking real values and an interval A R. The probability P(Sn 2 A) is said to follow a large deviation principle if there exists a real number GA, called the rate, such that nlim 1 ln P(Sn 2 A) = GA: (I.C.23)
The large deviation principle roughly means that the probability P(Sn 2 A) vanishes ex-ponentially fast with n: P(Sn 2 A) e nGA (I.C.24)
From this perspective it can be interpreted as a re nement of the law of large numbers, which states that the properly normalized sum of independent identically distributed random variables converges, with probability 1, to the common expectation value of the variables. Here, we observe that if GA vanishes, the probability measure concentrates, with probability 1, in the set A. In particular (in the case of the normalized sum of independent identically distributed random variables) if A is chosen to be any interval containing the expectation value of the variable, the large deviation principle provides the result of the law of large number together with the rate of convergence of the probability measure (which explains the denomination ’re nement’). This phenomenon is illustrated in the following example.
The Gartner-Ellis theorem provides a tool to compute large deviation functions through cumulant generating functions. It is commonly used in out-of-equilibrium statistical physics, especially to compute large deviation of particles current in the stationary state in interacting particles systems.

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#### Application to out-of-equilibrium statistical physics

We present brie y two relevant utilizations of the large deviation theory in the context of out-of-equilibrium statistical physics. This description is far from being exhaustive. We chose to focus on the applications of the theory which will be used later in this manuscript. The reader is invited to consult the review  for a complete description of the role of large deviation theory in statistical physics.
In out-of-equilibrium models, we saw previously that large deviation theory was the correct tool to deal with the uctuations of dynamic observables in the stationary state. It was indeed stressed that the uctuations of the dynamic observable were encoded in the quantity (I.C.13), which is exactly de ned as the cumulant generating function introduced in the large deviation context (I.C.32), where the time T plays the role of the large deviation parameter. The Legendre transformation then provides the large deviation function associated with this observable. By analogy with equilibrium models, the cumulant generating function may be related to a generalization of the free energy and the large deviation function may be connected to an entropy. These large deviation functions (or cumulant generating functions) are in general di cult to compute exactly: hence it is important to construct models for which such computations are possible. This motivates the study of integrable systems, i.e models for which analytical results can be derived. The next chapter is devoted to the presentation of integrability in the context of Markov chains. In chapter III we will compute the rst cumulants of the particle current of the dissipative symmetric simple exclusion process. We will explore in chapter IV the connection between the cumulant generating function of the current in the asymmetric simple exclusion process and the theory of symmetric polynomials.
A second application of the large deviation theory in out-of equilibrium physics, which will be of particular interest for us, is the macroscopic uctuation theory. This is a theory which aims to provide a coarse grained description of di usive systems. The probability to observe a path history, a time evolution, of properly de ned macroscopic variables satis es a large deviation principle. The large deviation parameter is the size of the system. The rate function can be heuristically interpreted as an action functional. The macroscopic uctuation theory provides also predictions about uctuations of the particle current and density in the stationary state. This theory will be presented in details in chapter V. The integrable models also play a central role in this context because they appear as benchmarks to test the predictions of the theory. We will indeed use in chapter V the exact results obtained in chapter III on the dissipative symmetric simple exclusion process and respectively on the open boundaries multi-species symmetric simple exclusion process to check the predictions for dissipative models and respectively to extend the theory to multi-species models.

I Out-of-equilibrium statistical physics
A Equilibrium versus non-equilibrium
1 Physical properties of equilibrium and non equilibrium systems
a) Equilibrium and entropy maximization
b) Non equilibrium and macroscopic currents
2 Dierent frameworks for out-of-equilibrium physics
B Markov chains and stationary states
1 Markov chains
a) Discrete time
b) Continuous time
c) Stationary state
2 Thermodynamic equilibrium and detailed balance
a) Time reversibility
b) Link with the Boltzmann distribution
3 Non-equilibrium stationary states
C Toward a description of NESS?
1 Generalized detailed balance and uctuation theorem
2 Dynamic observables and deformed Markov matrix
3 Gallavotti-Cohen symmetry and uctuation theorem
4 Large deviation functions
a) Large deviation principle
b) Legendre transformation and Gartner-Ellis theorem
c) Application to out-of-equilibrium statistical physics
II Integrability
A Introduction, motivations and formalism
1 Conserved quantities
a) Introduction and motivations
b) Markovian case
2 Exclusion processes framework
a) The conguration space
b) Markov matrix
B Integrability for periodic boundary conditions
1 R-matrix and transfer matrix
a) R-matrix and Yang-Baxter equation
b) Transfer matrix
2 How to nd R-matrices?
a) Direct resolution of the Yang-Baxter equation
b) Quantum groups
c) Baxterization
3 Diagonalization of the transfer matrix
a) Coordinate Bethe ansatz
b) Algebraic Bethe ansatz
c) Other methods
C Integrability for open systems
1 Reaction matrices and transfer matrix
a) K-matrices and reaction equation
b) Transfer matrix
2 How to nd K-matrices?
a) Direct resolution of the reaction equation
b) Baxterization
3 Diagonalization of the transfer matrix
III Matrix ansatz for non-equilibrium steady states
A Presentation of the method and link with integrability
1 General idea and example
a) General idea
b) Historical example of the TASEP
c) Pushing procedure for the open TASEP
2 Telescopic relations
a) Particular example of the TASEP
b) General case
c) Other examples
3 Thermodynamic equilibrium case
a) Some examples
b) A new look on the open TASEP stationary distribution
a) Algebraic setup
c) Ghoshal-Zamolodchikov relations
5 Ground state of the transfer matrix
a) Inhomogeneous ground state
b) Some examples
c) Inhomogeneous periodic TASEP
d) Inhomogeneous open TASEP
B Application to integrable models: examples
1 A diusive model with evaporation and condensation
a) Presentation of the model
b) Matrix ansatz
c) Computation of physical quantities
2 An open two species TASEP
a) Presentation of the model
b) Matrix ansatz
c) Identication
d) Explicit representation of the matrix ansatz algebra
e) Factorized form for the representation
f) Computation of the normalization from the explicit representation
3 An open multi-species SSEP
a) Presentation of the model
b) Markov matrix and integrability
c) Matrix ansatz
d) Computation of physical quantities
IV q-KZ equation and uctuations of the current
A Current counting deformation and q-KZ equation
1 Deformed Markov matrix
a) Denition of the model and Gallavotti-Cohen symmetry
b) Integrability of the deformed Markov matrix
2 Second deformation
a) Scattering matrices
b) q-KZ equation
B Koornwinder polynomials and link with q-KZ equation
1 Non-symmetric Koornwinder polynomials
a) Hecke algebra and Noumi representation
b) Non-symmetric Koornwinder polynomials
2 Symmetric Koornwinder polynomials
a) Finite dierence operator and symmetric Koornwinder polynomials
b) Link with the non-symmetric polynomials
3 Link with the q-KZ equation
a) Reformulation of the q-KZ equation
b) Reference state
C Matrix product solution to the q-KZ equation
1 Construction of the solution
a) General construction
b) Denition of the algebra
c) Construction of the solutions
d) Construction of the solutions to the left qKZ equation
2 Computation of Koornwinder polynomials
a) Normalization and symmetric Koornwinder polynomials
D Current uctuations conjecture
1 Quasi-classical limit
2 Generating function of the cumulants of the current
V Hydrodynamic limit
A Hydrodynamic limit and density prole
1 Hydrodynamic limit
a) Continuous limit of the lattice
b) Limit of observables in the DiSSEP
c) Limit of observables in 2-TASEP
d) Limit of observables in the multi-species SSEP
2 Large deviation functional of the density prole
a) Denition
b) Multi-species SSEP case
B Macroscopic
uctuation theory
1 Single species diusive systems
a) General idea
b) Stationary state
2 Single species systems with dissipation
a) Large deviation functional
b) Check with nite size lattice exact computations
3 Multi-species diusive systems: the multi-SSEP case
a) Large deviation functional
b) Check with nite size lattice exact computations
A Résumé en francais

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