Large deviations in nonequilibrium systems and phase transitions

Transcription

1 Licence / Master Science de la matière Stage École Normale Supérieure de Lyon Université Claude Bernard Lyon I M2 Physique Large deviations in nonequilibrium systems and phase transitions Abstract : In two non-equilibrium systems, the HMF model and Navier-Stokes, we observed phase transitions of second order whereas these systems have a phase transition of first order at equilibrium. The aim of my internship was to know if one could understand these phenomena with a perturbative computation of the rate functional of large deviations. The answer is negative, but the research on this question led to interesting results concerning the rate functional and instantons in stochastic systems. In the present report, we provide a small introduction to stochastic differential equations in statistical physics, and show why theory of large deviations is a useful tool when one tries to compute probabilities in stochastic systems. Freidlin-Wentzell s theory leads to introduce a variational problem, which is the minimization of an action over paths of the system. We then explain why the transverse decomposition of the vector field allows to solve the problem, and we apply this formalism to equilibrium problems in statistical physics. In the last section, we give a small view of nonequilibrium problems and how one can solve them perturbatively. Mots clefs : Large Deviations, Langevin dynamics, phase transition Stage encadré par : Freddy Bouchet Freddy.Bouchet@ens-lyon.fr / tél. (+33) Laboratoire de Physique, ENS de Lyon 29 juillet 214

2 Acknowledgements I would like to thank my advisor Freddy Bouchet who followed me and taught me most of the things I learnt during my internship. He gave me the direction of the research work I did and new ideas and encouragements when I didn t know how to proceed. He gave me a taste for research and I am very pleased to have been working with him for four months. I would like to thank Krzysztof Gawedzki, Eric van den Eijnden, and Cesare Nardini for valuable discussions in which I learnt much interesting physics. I am also very indebted to Tomas Tangarife who helped me when my advisor was absent, and gave me some advises to write this report. Jonathan Bertolaccini has been my coworker for all this time, I thank him for his very pleasant company, his calm and his attention. And of course this internship would not have been possible if the laboratory of physics of the ENS would not have accepted me to work. I know I was lucky to work in such a nice environment of scientists, and it gives me the desire to stay here for further works. Table des matières 1 Introduction 1 2 How is it possible to calculate a Large Deviation Functional? Langevin model, and stochastic differential equations The action in Freidlin-Wentzell s theory The variational problem and the Hamilton-Jacobi equations Transverse decomposition of the vector field Stationary measure in equilibrium models Models where the transverse decomposition of the vector field is trivial Equilibrium model for the two-dimensional Euler dynamics Some tools for solving Non-equilibrium situations Method to compute perturbativley the large deviation functional First-order correction in the Ginzburg-landau model and phase transition relationship between the vector field and the rate functional Perturbative approach in the stochastic Navier-Stokes equations Computation of the rate function on particular flows for the enstrophy measure Discussion about instantons Conclusions and perspectives 2 A Appendix. The one dimensional Langevin model in the underdamped limit 22 A.1 Large Deviations when ɛ goes to zero A.2 Stochastic averaging, and diffusion process for H B Appendix. Large deviations for the A-B model 24 C Appendix. Calculation of the rate functional in the Ginzburg-Landau model 27 D Appendix. Spatial correlations in the HMF model 28

3 1 Introduction Theory of large deviations has been taking an increasing importance into physics during the last thirty years, especially in the field of statistical physics. Mathematicians like Cramèr, Donsker, and Varadhan initiated this theory, and finally Freidlin and Wentzell in the 7 s [12] provided a complete rigorous framework and its application to stochastic differential equations. Although mathematically really challenging, (have a look in the book of Freidlin to be convinced of that), large deviation theory relies on simple intuitive ideas, and some of them were already known since a long time among the physicist s community. We say that a family of probabilities P n satisfies a large deviation principle with 1 rate α n and rate function f if lim n α n ln P n = f. Intuitively, this means that the probabilities P n look like exponentials for large n. In this report, I will sometimes use the notation P n exp (α n f) to say that P n satisfies a large deviation principle (LDP). In fact, one could argue that the development of equilibrium thermodynamics is the first use of large deviations into physics, and equivalence between thermodynamical ensemble is a typical example of a large deviation result. Lanford is to be credited for having brought the complete large deviation theory into the physicist s community, and his book [16] contributed much to extend its utilization for stochastic problems. When I discovered large deviations at the beginning of my internship, I was quite surprised not to have heard about it sooner during my studies, because all classical results in statistical mechanics can be regarded as applications of large deviations. A good review of large deviations and its applications into physics is done in [19], and I also found a nice pedagogical introduction from the mathematical side in [9]. One could wonder why large deviation theory is still relevant in statistical physics now that we have a satisfying comprehension of equilibrium thermodynamics. Now, physicists are mainly concerned by non-equilibrium problems, for which there is yet no general theory, because each system has its own specificity when submitted to a perturbation. In fact a lot of work is down to compute the probability of rare events in physics, that means an event whose probability is exponentially small. These events can be of a paramount importance because they can lead to qualitatively important changes in the system. A typical example is a system with two possible stable states, we will say two attractors. There is a small probability for it to cross an energy barrier thanks to fluctuations of its environment and go to a really different state. We will call it a non-equilibrium phase transition. Examples of bistable systems that display a non-equilibrium phase transition are to be found in 2D hydrodynamics. On earth, jet streams or also the Kuroshio -a strong current flowing along the coast of japan- can have two configurations and in previous years, scientists observed these currents going from one state to the other as shown on figure 1. There is already a review of such systems in [6] (see section 6). A rare event can also be a dangerous event, like an anoumalously fast spreading of pollutant gas. That s a motivation to study fast scalar dispersion as was done by J.Vanneste in [14, 15]. Large deviation techniques have proven to be very efficient for the study of long-range interacting systems where usual thermodynamics is not exactly valid. With long-range interactions, thermodynamical ensembles can be inequivalent and the computation of thermodynamical functions like entropy or free energy must be done separately. However, Bouchet, Dauxois, Barre, and Ruffo develop a technic using Ellis-Gärtner theorem to do these computations [1], and now, people have a good inview of ensemble inequivalence for long-range interacting systems. There are in fact very few systems where an explicit computation of the large deviation functional can be done, starting from a microscopic model. The first explicit computation might be the one done for the simple symmetric exclusion process by Derrida, Lebowitz, and Speer [11]. The problem took much time and work to solve, even if it is one of the simplest nonequilibrium situation one can imagine. This example show how challenging is for the physicist s community the problem of computing rate functions and therefore the probability of rare events. In this report I tried to give an overview of what I have learnt during an internship of 4 months. I could not report all I have done, I made some choices in order to keep a coherence and to avoid too many digressions that would have made my report quite messy. I focused mainly on bistable dynamics and on the question of non-equilibrium phase transitions. I wanted to bring the reader progressively to part (4) that deals with the real motivation of my work. The question was first to know if a phase 1

4 Figure 1 Kuroshio current. The system seems to have two attractors and the current changes from one to the other as shown on the time serie. Figure 2 Numerical simulation of two-dimensional stochastic Navier-Stokes dynamics in a box. The system has a bistability, one of the attractors is a dipole, the other is a parallel flow 2

5 transition could be modified when the noise has not its equilibrium structure. I therefore had to focus on systems where a phase transition occurs at equilibrium and I modified the noise by adding some spatial correlations. I hoped I could see some modifications with a perturbative computation of the rate function near the equilibrium situation. As the concepts I used in this part are not familiar to people who do not work in the field of statistical physics, I had to introduce step by step the tools related to stochastic differential equations (SDE). First I recall briefly the principle of an SDE and the large deviation principle (2.1), and I explain roughly how the problem of computing large deviations is equivalent to the minimization of some action functional. Then I introduce some simple models with an SDE, and show how one can solve easily the equilibrium situation (3). The last part presents the theory of perturbations on rate functions, which was first initiated by my own director F.Bouchet with K.Gawedski, and C.Nardini, and an application to the stochastic Ginzburg-Landau equation. The latter calculation led us to formulate a small theorem on the link between rate functions and the dynamics of the system. In the Hamiltonian Mean Field (HMF) model, which is a simple classical model of many particles interacting through long-range interactions [8], with a noise out of equilibrium, numerical simulations have shown that the system has a first order phase transition, whereas the same system at equilibrium (i.e with a noise white in space and time) has a second order phase transition (6). The motivation of my internship was to know if one could explain this observation with the perturbative computation of the rate functional that gives the stationary probability of the system. To answer this question, I first had to learn the stochastic calculus and the Itô convention, then the Freidlin-Wentzell theory that allows to compute the quasipotential for systems with small noise. This took me a lot of time, because the Freidlin-Wentzell theory is not taught at the university and I also wanted to understand the mathematics related to large deviation theory, the Ellis-Gärtner theorem and convex analysis. Although it does not appear in my report, I had to know about the computation of large deviations in slow-fast dynamical systems, which uses the time scale separation. Last, I had to learn the perturbative computation of a rate functional. The bibliographic part of my internship took me about two months, before I was ready to do calculations for complex systems. As a kind of tutorial for my internship, I discussed the commutation of the small noise limit and small friction limit in the one-dimensional Langevin model. I present the results in appendix A. Afterwards, I derived the equation for the empirical density in the HMF model, both at equilibrium and out of equilibrium. But I noticed that the structure of the stochastic term is very different in those two situations. In the out of equilibrium model, the scalar product has a degenerescence, and hence this model does not seem at all to be a perturbation of the equilibrium situation. We didn t know if this kind of model is relevant from a physical point of view, we therefore chose not to treat the difficulties that are specific of this model, but I rather studied first a simpler dynamics, the stochastic Ginzburg-Landau equation. With some calculations in case of the Ginzburg-Landau dynamics with small out of equilibrium noise, I showed that the phase transition is not modified, contrary to what we expected. In fact, we found a more general argument presented in section (4) to understand why a phase transition cannot be affected by the structure of the stochastic term. This argument provides a relation between the stationary points of the vector field and the extrema of the rate functional. After we saw this argument, I turned to the two-dimensional Navier-stokes dynamics, and proceed in the perturbative computation of rate functionals. The latter computations were motivated by numerical simulations where one observes a bistability for two-dimensional flows in a box (2). The equilibrium theory of two-dimensional Euler dynamics [3, 6] shows that a phase transition of second order can occur between a dipole and a parallel flow. The simulations, which were done out of equilibrium, show phase transitions with the same attractors. However the out of equilibrium transition is of first order, and we do not know how to make the link between equilibrium and non-equilibrium. This example reminds what happens for the HMF model. This system is A possible attempt to understand this phenomena was to compute the large deviation functional in case of a non-equilibrium noise to see if the two observed flows are minima of the rate functional, or at least if the rate functional is smaller on these two flows than on any others. I studied as a simpler model the AB model to understand general 3

6 features of the rate function that could be valid in case of the Navier-Stokes dynamics, and I did some conjectures that could be the starting point for further works. 2 How is it possible to calculate a Large Deviation Functional? To find the stationary probability of a system for which a large deviation principle for this probability law exists, we need to compute the rate functional in the large deviation principle. The rate functional gives the probability of states for the system, and hence we can see if there is any phase transition in the system. In this section we present the fundamental tools for the computation of large deviations in stochastic models. We first recall the definition of the large deviation principle, and we give next the results of Freidlin-Wentzell theory we will use all over this report. We show how the computation of a rate function consists in minimizing the Freidlin-Wentzell action, and we end up this section with the definition and utility of the transverse decomposition of the vector field. 2.1 Langevin model, and stochastic differential equations In statistical physics, we often have to consider systems with a very large number of degrees of freedom. A typical example is the case of a particle immersed in a surrounding bath of other smaller particles. One could hardly describe the motion of all particles, because their number is about N A, the Avogadro constant. Moreover, we are not interested in the motion of all particles but only in the first larger one. A good idea is then to forget the motion of particles in the bath and to model all interactions of the particle with the bath as a random force applied on this particle. This type of model is called a Langevin model. In that case, the equation that describes the motion of the particle with position X and momentum V is { dv = du dx dt νv dt + 2νɛdW (1) dx = V dt. Here we consider that the particle is submitted to an external force that is the gradient of a potential. The hamiltonian of this system therefore is H(X, V ) = V U. This is the example of a stochastic differential equation. (all over this paper, I follow ItÙ convention for stochastic equations) ɛ is the nondimensional temperature, and ν is the friction coefficient. Interactions with the surrounding environment are all contained in the stochastic term dw, which is also called in mathematics a Wiener process. This is a random small increment that follows a gaussian law with variance dt. This means that its probability law writes P (dw ) exp ( dw 2 2dt A careful reader would certainly notice that the friction coefficient ν also appear in the noise intensity. Indeed, the noise is linked to dissipation with a fluctuation-dissipation relation. With this relation, the probability law of the particle in phase space is Boltzman s measure P (X, V ) ( exp H(X,V ) ɛ ) This measure solves the Fokker-Planck equation associated with the Langevin equation. Although we do not know the exact trajectory of the particle, we certainly have already said much about this problem. Large Deviation Principle for the probability law P (X) The marginal law associated with Boltzman s measure above is P (X) exp ( U ) ɛ, with the assumption that min U(x) =. Then it is straightforward that lim ɛ ln (P (X)) = U(X) ɛ ). 4

7 We say that the probability law follows a large deviation principle with rate 1/ɛ and rate function U(x). As a consequence, when ɛ goes to, the particle can hardly reach a region where the potential is high. Take a potential U with two minima separated by a saddle point : when the noise intensity goes to zero, the particle will almost never quit the basin of attraction of the minimum it is inside, because of the barrier of potential. However, there is a small probability for this to happen, and it could indeed happen over a very long time : we will call this a rare event. Large deviation theory aims to compute the probability of such an event, and to answer another important question : which of all possible trajectories the particle will follow to escape its basin of attraction? Are all trajectories equivalent, or is one trajectory much likely to happen? The trajectory of the particle is of course random, but if the path probability follows a large deviation principle (LDP), one trajectory is much likely to happen. The probability concentrates near this particular trajectory. We will call it an instanton. The following statement acts as a summary for large deviation theory : Among all possible rare events, this is the less rare that effectively happens. Freidlin-Wentzell s theory has already justified rigorously this idea. But in this paper, I do not want to demonstrate the relations I will use, I am not able to do that anyway 2.2 The action in Freidlin-Wentzell s theory I will now consider a more general case than the one I discussed in equation (1), because the system will not be of finite dimension. In many interesting physical models, the variable is a field, so we need a theory also valid for these systems. I write an SDE (stochastic differential equation) for a field f(x, t), which can be a scalar field- for example a concentration profile- or a vector field- for exemple a velocity profile in a flow. I say that f obeys an SDE that writes f t (x, t) = H(f) + 2ɛη(x, t). (2) One has to think about H as a differential operator (as in Navier-Stokes equations or in advectiondiffusion equations), and I will often refer to this term as the vector field with analogy to differential equations. ɛ is a measure of the noise intensity, the noise is now written η. It is no longer a random variable, it is a random field parameterized by x. One can think of a gaussian vector with infinite size to have an idea about it. The correlation function of the noise is E(η(x, t)η(x, t )) = C(x, x )δ(t t ). The probability law of the random field is derived simply by considering the finite-dimensional case, and extend it to an integral over fields. In the finite-dimensional case, the probability of a gaussian ( vector would be P [η] exp 1 2 ηt i C 1 i,j η j components, the matrix product becomes an integral over fields ( P [η] exp 1 ) dxdx C 1 (x, x )η(x, t)η(x, t ) with 2 ), where η is a vector and C 1 a matrix. With an infinity of dx C(x, x )C 1 (x, x ) = δ(x x ). (3) With this formalism, the probability law of a field f(x, t) with value f at time zero is obtained by addition of all small random increments. Consider the case where f is an attractor of the dynamical operator H, then it is clear that the system can reach another statef 1 only within a time very large compared to the typical relaxation time of the dynamics. The system has a greater probability to reach f 1 as the time increases. We are then interested in the limit probability for the system to reach f 1 when time goes to infinity. Starting from equation 3, we do a change of variable to remove η, using equation 2. The new variable is f. The probability to reach f 1 is a sum of probabilities of all possible paths starting from f and ending at f 1 with infinite duration. I don t give a rigorous proof of this expression, I just made it believable. It writes ( P [f 1 ] = N D [f] exp 1 f 4ɛ t H f ) t H (4) f 1 f 2 = dxdx C 1 (x, x )f 1 (x)f 2 (x ), (5) 5

8 where N is some normalization constant. One has to pay attention to something : in the expressions above, there is a quadratic form in a functional space, and it is a scalar product if the correlation function of the noise has good properties. Take for example a noise δ correlated in space, you see that C 1 (x, x ) = δ(x x ) and that the scalar product is no more than the standard one in L 2. One can also define a gradient associated with this scalar product. The gradient of a functional F is defined through the relation δf[f] = gradf δf. As we also have with the usual relation δf[f] = dx δf δf (x)δf(x), the general expression of the gradient associated with the scalar product is gradf(x) = dx C(x x δf ) δf(x ). In the limit of small noise, ɛ, one immediately sees that the integral is dominated by the minimum of the factor in the exponential. The Laplace principle allows us to give an equivalent of the probability in 4 when ɛ goes to zero. We define by analogy with mechanics the action and a pseudo-potential A(f, t) = 1 4 t f t f H(f) t H(f) F(f 1 ) = lim t min {A(f, t) f(., t) = f ; f(., ) = f 1 }., (6) Following Freidlin-Wentzell s theory, the probability law P(f 1 ) satisfies ( the large ) deviation principle with a rate F an a rate 1/ɛ, this means heuristically that P(f 1 ) exp. This result is a kind F(f 1) ɛ of generalization of the Laplace method used in physics, although here integrals are performed over paths and not over R. We take the minimum of the factor in the exponential- here the minimum of the action-, and this gives the equivalent of the probability law up to some algebraic corrections. It may not be obvious, but P(f 1 ) is also the stationary probability of the system, which is an important fact. It implies that if we are able to find the stationary probability, i.e we solve the Fokker-Planck equation, we know the rate functional. I do not give the proof of this fact, but I wanted to put attention to that point because I will use it in the resolution of the HMF equilibrium model. The trajectory f(x, t) that minimizes the above defined action is called the instanton. The difficulty lies in solving a variational problem in order to find the rate functional and the instanton that corresponds to this minimum. The computation of rare events is the motivation for this task. If the system is trapped in the vicinity of an equilibrium point of the dynamics, what is the probability that it goes far from this point? Large deviation theory states that this probability is exponentially small, and it gives the dominant factor inside the exponential. Now we know that large deviation theory is useful to compute the probability of rare events. The probability of such events concentrates around one single trajectory called the instanton. The instanton is the trajectory that solves the problem of minimizing the action with given boundary conditions. How to compute this minimizer and the minimal action is in itself a difficult task. We will see a possible method in the following section. 2.3 The variational problem and the Hamilton-Jacobi equations In classical Mechanics, one often has to find the minimizer of an action, that s why there are already many techniques to solve such a problem. The most popular one consists in writing Euler-Lagrange equations and solving them, because they give the real trajectory followed by the system. Usually, people are only interested in trajectories, and they give some initial conditions to solve Euler-Lagrange equations. Differential equations with given initial conditions is called a Cauchy problem, it has a unique solution. In classical mechanics, the action is simply a trick to rewrite the trajectory as the minimizer of some convex functional. Here this is not the case, because the action gives the pseudopotential, and that s exactly the quantity we want, we want the rate functional. It would be useless to 6

9 solve Euler-Lagrange equations and compute the rate function by inserting the result in the expression of the action. Moreover, we do not give initial conditions to solve the problem, we impose an initial condition and a final condition. The existence of a trajectory is not at all obvious, neither is the uniqueness of the solution. There is another way : Hamilton-Jacobi equations, which give immediately the minimal action, provided we are able to solve a PDE of order one. Hamilton-Jacobi equations for the variational problem in classical mechanics formally writes (for an introduction to Hamilton-Jacobi equations, see the course of classical mechanics by L.D. Landau & E.M. Lifshitz) : ( F (x, t) + H x, F ) t x, t =. Here F is the minimal action (in classical mechanics, it is usually called S), it depends on the final coordinate and time, H is the hamiltonian. We will compute in our particular case, given the action (6), the hamiltonian. First we need the conjugate momentum associated with f, we call it p. L = 1 4 < f t H f t H > is the Lagrangian, (7) ( ) p = grad f L = 1 f 2 t H is the conjugate momentum. (8) t That gives us the following Hamiltonian : H = p f t L = p p H. One has to be careful about that : momentum, Hamiltonian are defined with the good scalar product, the one that appears in the action (6). The above expressions are a generalization of the ones that are found in classical mechanics books, where the scalar product is implicitly the one in L 2. Here we are dealing with an Hamiltonian where time do not appear explicitly. As a consequence of this fact, the Hamiltonian is constant along trajectories, and this constant is zero because it starts from an equilibrium point of the vector field (see the expression of the momentum above). The Hamilton- Jacobi equation (2.3) then simplifies in H (f, grad f F) =. Given the expression of the Hamiltonian, we find grad f F grad f F = grad f F H. (9) Expression (9) is the main equation in this section. The solution of this equation is F, the rate functional, which is the minimizer of Freidlin-Wentzell action. The problem is that no general method exists to solve such a PDE of order one, that can possibly contain nonlinear terms. 2.4 Transverse decomposition of the vector field Writing the hamilton-jacobi equations (9) did not bring us much further into the resolution. In general, such equations do not have any simple solutions. However, let s continue to rewrite the problem in another way. Have a look at equation (9) : we can rewrite it gradf G = where we defined G def = gradf H. We may solve the problem if we find some functional F such that its gradient is normal to the quantity G. Let s say that another way : if there is any chance that we can separate the vector field in two parts, one that writes gradf and another one G normal to the first one, the problem becomes trivial. We are then sure that F is the rate functional we are looking for. We also wanted to know the equations of the instanton. If we find the rate functional F, it is possible to obtain these equations without computing the Euler-Lagrange equations. We know from classical mechanics 7

10 that p = grad f F. But we also calculated the formal expression of p in the equation (8). Equating both expressions gives the equations of the minimizer of the action, namely the instanton, f t = gradf + G. It would be really surprising that such a decomposition appears in our equation, but it s indeed what sometimes happens! In most Hamiltonian systems with some dissipation, the dissipative terms are orthogonal to the conservative dynamics for some scalar product. It is then sufficient to introduce a noise that gives this particular scalar product. Some examples may illustrate this point : example : the model A-B The model A-B has little physical interest, but it is worth studying it from a theoretical point of view. First, it is one of the simplest model one can imagine, second I was studying it to understand some generic properties of perturbed Hamiltonian dynamics, especially the two-dimensional Navier-Stokes equations. We will return to this model later in the report. It is governed by the equations [5] { da = ABdt νadt + νɛdw A db = A 2 dt νbdt + νɛdw B. AB and A 2 is the conservative part of the dynamics, whereas νa and νb is the dissipative part. It is constructed with a dynamics that conserves the function f(a, B) = 1 ( 2 A 2 + B 2). The two Wiener processes dw A and dw B are independent, ( and ) the scalar product is therefore the standard product in R 2. One will then writes gradf = f A ; f B = (A; B) The dissipative part is obviously the gradient of the function f and a straightforward computation shows that it is also orthogonal to the conservative part. We conclude that f is the rate function with the rate 1/ɛ. To obtain the equations of the instantons, just reverse the sign of the dissipative terms : da dt db dt = AB + νa = A 2 + νb 3 Stationary measure in equilibrium models In the first part, we saw how to compute a rate functional by solving a variational problem. Then we have rewritten this problem in terms of the orthogonal decomposition of the vector field, thanks to Hamilton-Jacobi equations. Of course this is absolutely not trivial to find this decomposition, unless it appears obviously as a conservative dynamics and a dissipation. In fact, it sometimes happens that we can find the rate functional, because the stationary measure of the problem is known : If one can solve the Fokker-Planck equation associated with the stochastic model, then the solution gives the rate functional. I will give in this section a few examples of such problems where an orthogonal decomposition of the vector field is obvious. 3.1 Models where the transverse decomposition of the vector field is trivial exemple 1 : the stochastic Ginzburg-Landau equation The Landau modelization of phase transitions is well known in physics : it is a mean field approach where one develops the free energy in terms of the order parameter f in the vicinity of the critical point. Usually, this development respects the symmetry f f, and in that case only terms that respect this symmetry appear. With a f 4 development of the local potential U, one can model a second 8

11 Figure 3 Trajectories in the model A-B without dissipation. The red line represents the ensembles of stable stationary points, the unstable ones are on the blue line order phase transition, whereas with a f 6 development, a first order transition occurs. I was studying the Ginzburg-Landau model because it seems to be the simplest model with an infinite number of degrees of freedom, and it has a phase transition at equilibrium. Moreover the stochastic part added to the G-L equations 1 below is much simpler than the one I found in the HMF model. I thought that the G-L model then was a first step to understand generic features of a stochastic system near a phase transition. The free energy will be then F[f] = dx [ 1 2 f 2 +U(f) ]. But this expression is not sufficient to understand how the field f relaxes towards its equilibrium value, which is given by the minimum of the free energy. People then proposed some phenomenological equations to simulate this relaxation, but the equation where the order parameter is conserved must be different from the one if it is not. In case of a non-conserved order parameter, the Ginzburg-Landau (G-L) equation writes f t = δf δf = f U f. This equation makes sure that the system relaxes towards the minimum of F, because the free energy is a Lyapunov functional of the equation above. However, this relaxation is fully deterministic and does not take into account the fluctuations one can usually observe in a real system. For example, if F has two minima, what occurs in a phase transition, the system may be able to switch from one minimum to the other. Let s add a stochastic term in the G-L equations, to obtain f t = δf δf + 2ɛη(x, t), (1) E[η(x, t)η(x, t )] = δ(x x )δ(t t ). Usually, ɛ is a temperature, because fluctuations in a real system come from the fact that the temperature is non-zero. It is worth to notice that equation (1) was really challenging for mathematicians 9

12 and physicists, because it is not obvious how we can give a meaning to it. This has been done in ([2]), but I will not enter into these subtleties. The noise was chosen to be δ correlated in space, in order to make the scalar product as simple as possible. In that case, the term δf δf is already the gradient of the free energy F, and there is no transverse dynamics, G is simply zero. We conclude that the free energy is the rate functional, and that the equation that describes the dynamics of the instanton is f t = δf δf, and f starts from a minimum of the free energy. One remark is important here : if the temperature is under the critical temperature, there is a phase transition and the free energy has two minima. In F-W (Freidlin-Wentzell) theory, a pseudo-potential is uniquely defined in the basin of attraction of a stationary point of the dynamics,i.e a point where the vector field vanishes. When we compute a rate functional, it is valid only in the basin of attraction of a stationary point. The stationary probability we will find is not defined in the whole space of parameters, but only in this basin of attraction. We obtain the probability to reach the saddle point. If the system crosses the saddle point, it will relax fast to the other stationary point following the dynamics f t = δf δf. In the other basin of attraction, we can also calculate the rate function, and the stationary probability. It is not at all obvious that a unique smooth rate function can be globally defined. In fact, Freidlin and Wentzell already studied the case of many attractors, and the question to know if the rate function is global. This question has to be investigated further. We already know examples where the rate functional is not smooth over space. The curious reader could read the paper of Graham [13] exemple 2 : The Hamiltonian Mean Field (HMF) model The HMF model has already been throughout studied because it is a simple example of manyparticles dynamics, with long range interactions. The equilibrium problem can be solved analytically, one can explicitly compute the entropy, or the free energy depending which of the thermodynamical ensemble is chosen for the computation, but both ensembles were shown to be equivalent. For me it was particularly interesting because the model displays a phase transition of second order under the critical temperature T = 1 2. I wanted to study perturbatively the non-equilibrium case because I thought the phase transition could be modified, and become first order. This assumption seemed to be supported by numerical simulations [18].For a review of the results on this model, see e.g [8]. The important point for our purpose is that there is a phase transition of second order, that s why it is worth studying this model from the point of view of Large deviations. Is it possible to compute the instanton that makes the system switch between two attractors? The HMF model is a Langevin dynamics of N particles with white noise applied to them : dθ i = p i (11) dt dp i = dv dt dθ (θ i) αp i + 2αk B T η i (12) Here α is a friction coefficient, the η i are independent white noises and V (θ) = 1 N w(θ θ i ) is a mean field potential that describes interactions between particles. Interactions are therefore long-range, because the contribution of particle i does not depends on its distance to the point θ. Particles are localized on a circle, and θ varies from to π. We are in fact interested in the empirical density that writes f N (θ, p, t) = 1 N δ(θ θi (t), p p i (t)). This expression measures the density of particles at time t and at (θ, p) in phase space. It is a good variable to look at, in order to see any macroscopic changes in the system. With Itô calculus, it is straightforward to derive the stochastic equation of the empirical density f N t + θ (pf N) p ( ) dv dθ f N = α ( pf N + k B T f ) N + 2αkB T p p p N f Nη(θ, p, t). (13) i 1

13 k B T N is a natural parameter that appears as a rate for large deviation in front of the noise k BT N. The noise η is white in time and space. Given the equation above, it seems very tricky to apply our formalism and to try to compute a rate functional, and instantons. However, everything will turn out fine because we are dealing with a Langevin model. First of all, it is not that much complex to derive the stationary measure for f N. To see this, we have to compute whithin the canonical ensemble the probability law. Given that each configuration appears with a weight proportional to e N k B T HN, the probability law is P S [f N = f] = {dθ i dp i } δ(f N f)e N k B T HN. H N is the sum of the energies of all particles, H N = p 2 i 2 + w(θ i θ j ), and can be rewritten as i (i,j) H N = 1 p 2 f N + 1 f N (f N w). 2 2 The star stands for the convolution product f w(θ) = dθ dp f(θ, p )w(θ θ ). We then apply Sanov s theorem to evaluate the quantity {dθ i dp i } δ(f N f), which is no more than the microcanonical measure of the distribution f. We have {dθ i dp i } δ(f N f) e f ln f. (for an explanation of Sanov s theorem, see i.e [9]). We find that the stationary measure satisfies a large deviation principle with rate functional F and rate N k B T P S [f N = f] e N k B T F[f] with the free energy F[f] = 1 2, what means exactly that p 2 f + 1 f.f w + k B T 2 f ln f. (14) I recall here that the symbol was defined in the introduction and means that the probability satisfies a LDP. The reader familiar with thermodynamics will easily recognize a free energy as the sum of an energy and an entropic term. We expect it to be the rate functional in F-W theory, because the two computations - one within the canonical ensemble, the other with F-W theory- should be consistent. I explained this point in section 2.2. To be convinced of that point, let s now compute the scalar product associated with noise and the orthogonal decomposition of the vector field. The scalar product is somewhat complicated, because it depends explicitly on the distribution f. Take equation 13, the correlation function of the noise is E [ p ( fn (θ, p)η(θ, p, t) ) p ( fn (θ, p )η(θ, p, t ) = p p f N (θ, p, t)f N (θ, p )E [ η(θ, p, t)η(θ, p, t ) ] }{{} =δ(θ θ,p p )δ(t t ) = ( fn (θ, p, t)δ(θ θ, p p )δ(t t ) ). p It means that the correlation function is C(θ θ, p p ) = p (f N(θ, p, t)δ(θ θ, p p )). To obtain the scalar product, just use the formula (4). Given one distribution f, define r 1 r 2 f = dθdpj 1 (θ, p)r 2 (θ, p) with j i defined through r i (θ, p) = p f(θ, p) p j i(θ, p). and the quasi-potential writes explicitly { } A[f ] = min dt < t f H[f] t f H[f] > f f( inf) = f S ; f() = f. In this expression, f S is a stable stationary solution of the vector field H, which is the big operator acting on f in equation (13). It is easy to check that the quadratic form defined above is a scalar product, it is symmetric and strictly positive. Our point is to show that A defined above is in fact the free energy F. We proceed as follow : 11 ) ]

14 We first give the expression of the gradient of the free energy, and check that it is indeed the dissipative part of H Then we verify that the rest of the operator H (the vlasov-part of equation (13), we will call it G) is transverse to the dissipative part. This will prove that we have a transverse decomposition of the vector field, and following what we said in part (2.4), we will solve the problem. Let s go : Given the form of the scalar product, the gradient of F is gradf = p f δf p δf. Then take the ( ) expression of the free energy given in (14) an you find exactly gradf = p pf N + k B T f N p. The last step is to compute < G gradf > f = dθdp G δf δf [ = dθdp (pf) + ] [ ] p 2 θ p ((w f).f) 2 + (w f) + k BT lnf + k B T (integration by part) = dθdp.pf [ ] p 2 θ 2 + (w f) + k BT ln f + k B T dθdp((w f).f) [ ] p 2 p 2 + (w f) + k BT ln f + k B T = dθdp k B T p f θ dθdp k B T (w f) f p (by integration of each term) = And the proof is complete. We end up this section with the equations of the instanton, which are readily f N + t θ (pf N) ( ) dv p dθ f N = α p le signe change! f(., ) = f S. ( pf N + k B T f N p The HMF model is a Langevin model, i.e a type of model where the dissipation is related to the stochastic term exactly in such a way that the transverse decomposition is trivial. In fact, the microscopic model (12) has already a trivial transverse decomposition. The calculus above shows that this transverse decomposition is preserved when one derives the stochastic equation for the empirical density. 3.2 Equilibrium model for the two-dimensional Euler dynamics In previous works, (see the paper [4]) some people have noticed that a phase transition can occur in a stochastic flow modeled by Navier-Stokes (NS) equations in two dimensions in which a stochastic term is added. In a box with periodic boundary conditions, those people observed in numerical simulations that the flow can organize itself at a macroscopic scale into a unidirectional flow or into a dipoles. Both flows are stationary solutions of the Euler equations, and numerical simulations show that they are attractors of the system. The transition between these two attractors still has to be understood. To achieve this aim, it is worth to understand first the properties of the SNS (stochastic Navier-Stokes) equations and to see if we can use the techniques of large deviations in such a system. Taking the curl of Euler equations in 2D gives ω t + V. ω =. We use the notation ω for the curl of the velocity V. This is an Hamiltonian system, with an infinity of conserved quantities, the Casimirs and the kinetic energy. So far there is no dissipation term. Let s 12 ),

15 add in this equation a stochastic force of the form νɛη(x, t). ν will be the dynamical viscosity, and ɛ a small parameter for Large deviations. The random field η has the correlation function C(x x ) in space and is white in time. if we want the vector field to have a transverse decomposition as G gradf, we must add a dissipative term in the Euler equations. The gradient of a functional F is gradf[ω] = C δf δω. We can then write the following equation : ω t + V. ω = ν C δf δω + νɛη(x, t). (15) The above equation is a stochastic model for 2D Euler-equations. The functional F is the rate functional in the Large deviation principle iff both terms V. ω and C δf δω are orthogonal for the scalar product associated with the noise, i.e iff dx δf δω V. ω =. This means exactly that F should be a conserved quantity for Euler dynamics. We can then take one of the casimirs to build our equilibrium model. However, only the choice F = 1 2 dx ω 2, which is called enstrophy and C(x x ) = δ(x x ) can give the term of dissipation in NS equations. The SNS equation writes then ω t + V. ω = ν ω + νɛη(x, t). (16) One should notice something : this equation has no straightforward mathematical meaning. As in the case of Ginzburg-Landau equations (1), some quantities are diverging, and ω is no longer a function but a distribution. This can be seen when one writes the energy balance, and it appears that the power injected by the noise is infinite. A product of two distributions has no meaning yet. Freidlin ([7]) tried to give some meaning to equation (16) by taking a smooth correlation function for the noise C l depending on the small scale l, such that C l (x x ) δ(x x ). He proved then that for l finite l, the rate function is well defined and that it converges to the enstrophy when l goes to zero, as was expected from a formal computation in SNS equations. The regularization process is also used for numerical simulations, because it is impossible to do computations with a noise white in space. However, numerical simulations do not at all use a small l, they use a finite l, sometimes not even small compared to the size of the box. Therefore in numerical simulations like the ones done in [4], the rate functional of large deviations is no longer the enstrophy. In our case there is only one attractor which is ω = for the dynamics, and which is also the minimum of the rate function Enstrophy, so we cannot really hope to explain numerical simulations with a perturbative approach like the one we will do in the next section. To sum up, the two-dimensional stochastic Navier-Stokes equation is a very interesting system : we can solve explicitly the equilibrium case thanks to the trivial transverse decomposition, and the non-equilibrium case has a physical importance to understand turbulence. This model is not just a "study-case", it is worth considering it with a physical point of view. 4 Some tools for solving Non-equilibrium situations In section 2, we have seen that some stochastic models can be solved exactly in the case where the noise is white in space. In these models, the transverse decomposition of the vector field appears trivially as the conservative terms and the dissipative terms. But we are interested in real systems or in numerical simulations where the noise is not at all white in space, but whose Fourier transform is zero under a scale l. That means that Fourier coefficients of the noise are all zero under some cut-off of size l. We will say that these models are non-equilibrium models because the rate functional is not the one given by equilibrium thermodynamics. In case where noise is not white in space, the trivial transverse decomposition breaks down, and one possible approach consists in computing perturbatively the rate functional when the correlation size goes to zero. The motivation of my work at the beginning was to see whether a system displaying a phase transition at equilibrium would be qualitatively modified when correlations are added. I asked myself if the phase transition could change its order or its localization in the space of parameters. These questions led me to investigate the link between the vector field and the rate function of large deviations. 13

16 4.1 Method to compute perturbativley the large deviation functional In this section, we will see that we can develop the rate function in the correlation length l, and compute order by order the terms appearing in the development. Our starting point is a SDE of the form : f t = grad F + G + ɛη l (x, t), with the correlation function E[η(x, t)η(x, t )] = C l (x, x )δ(t t ) where C l (x x ) l C (x x ). The gradient is defined by the noise, grad F [f] = C δf δf, and F is the rate functional when l =, which means that < gradf G > =. I put the subscript to emphasize that gradients and scalar products are related to l, and here l =. For l = we can solve the problem, we know a LDP which is P [f] e F [f] ɛ, we know the instanton. l = is the equilibrium situation in problems like HMF or Ginzburg-Landau. But now l is no longer zero, we are left with a non-equilibrium situation. The problem is that the scalar product changes because grad l F [f] = C l δf δf f t = grad lf + G + ɛη l (x, t),. Then we could rewrite 4.1 as with a new definition G := G + grad l F grad F. In non-equilibrium dynamics, we do not know a priori the transverse decomposition and there is absolutely no reasons why grad l F and G should be orthogonal. That s why F is not a priori the rate functional. And yet we have almost a transverse decomposition because < grad l F G > l=< grad F G > =< grad F grad l F grad F >. The latter quantity is certainly of order 1 in l because grad l is just a perturbation of grad. We can then write < grad l F G > l= lt 1 [f]. The rate functional will certainly write F = F 1 + O(l 2 ) and our aim is to compute F 1 first. To be more precise, there exists certainly G = G 1 + O(l 2 ) such that < F 1 G 1 > is of order two in l and gradf + G = gradf 1 + G 1. I will not report here the general demonstration, which has be done by F.Bouchet and K.Gawedski, and will soon be published. I will rather write the expression for F 1 and compute it in our particular case of a modified noise. The general theory give for F 1 the expression F 1 [f] = F [f] l dt T 1 [R (t, f)], where the path R (t, f) is defined through the equation : { t R = grad l F [R ] + G [R ] R (, f) = f is the initial condition. I assume that the new correlation function C l has an expansion in distribution space as C l (x x ) = C (x x ) + ld 1 (x x ) + O(l 2 ). Then I want to compute the functional T 1. We write simply < grad l F G > = dx δf [ (C l C ) δf ] (x) δf(x) δf = l dx δf ( D 1 δf ) (x). δf(x) δf The first correction of the rate functional is therefore F 1 [f] := l dt T 1 [R (t, f)] = l dt dxdx D 1 (x x ) δf δf [R (t, f)] (x) δf [ R (t, f) ] (x ). δf (17) Let s comment a bit on this expression : The correction for the rate functional requires to integrate the characteristics of the fluctuation dynamics (I mean the dynamics where the dissipative part is 14