Revisiting classical and quantum disordered systems from the unifying perspective of large deviations

Abstract

The theory of large deviations is already the natural language for the statistical physics of equilibrium and non-equilibrium. In the field of disordered systems, the analysis via large deviations is even more useful to describe within a unified perspective the typical events and the rare events that occur on various scales. In the present pedagogical introduction, we revisit various emblematic classical and quantum disordered systems in order to highlight the common underlying mechanisms from the point of view of large deviations. I. INTRODUCTIONJust like Mr Jourdain discovering that he has been speaking in prose all his life without knowing it, physicists working in statistical physics become aware at some point that they have been using the theory of large deviations without realizing it since their very first acquaintance with the Boltzmann notion of entropy and the Gibbs theory of ensembles. This language of large deviations has turned out to be very powerful to unify the statistical physics of equilibrium, non-equilibrium and dynamical systems (see the reviews [1-3] and references therein) and to formulate an appropriate statistical physics approach of dynamical trajectories for various Markovian processes (see the reviews [4-10] and the PhD Theses [11][12][13][14] and the HDR Thesis [15]).In the field of disordered systems, the presence of random disorder variables induce a lot of subtle effects for the probabilities of interesting observables. Physicists have understood from the very beginning that some observables are non-self-averaging, i.e. their disorder-averaged value is completely different from their typical value (see the books [16,17] and references therein). It was also realized very early that in each large typical sample, there will nevertheless occur rare anomalous regions of a certain size that may dominate some observables : famous examples are the Lifshitz essential singularities of the density of states near spectrum edges in Anderson localization models [16,[18][19][20][21], the Griffiths singularities for the statics [22,23] and the dynamics [24-26] of random classical models, and the Griffiths phases in random quantum models (see the reviews [27,28] and references therein). Finally at critical points, it was found that multifractal properties appear, for instance for the inverse participation ratios of eigenfunctions at Anderson localization transitions (see the reviews [29, 30] and references therein) or for correlation functions in random classical spin models [31][32][33][34][35][36][37][38], while at Infinite Disorder fixed points, many observables are even more broadly distributed [27,28]. These few examples indicate that that the language of large deviations is even more useful in the presence of disorder in order to describe within a unified perspective all these phenomena involving typical and rare events on various scales.The aim of the present pedagogical introduction is thus to explain to physicists how the general theory of large deviations is the natural language to analyz...

Role of current fluctuations in nonreversible samplers

“…Our approach consists in adding current fluctuations in the analysis, using the level 2.5 of large deviations, in order to get a clearer and more physical picture of why nonreversible samplers have better convergence properties statistically compared to reversible ones. This provides an application of the level 2.5 in statistical estimation, complementing the more physical applications that have been discussed up to now [41][42][43][44][60][61][62].…”

Conditioning two diffusion processes with respect to their first-encounter properties

“…When the single diffusion of Eq. 4 corresponds to the linear restoring drift towards the origin x = 0 with the parameter k > 0 µ(x) = −kx (108) and to the diffusion coefficient D(x) = 1/2, the 1-particle Ornstein-Uhlenbeck propagator…”

“…Within the classification into levels of the large deviation theory (see the reviews [94][95][96] and references therein), the Level 2 concerning the empirical density alone is usually not closed for non-equilibrium processes that do not satisfy detailed balance. However, the Level 2.5 concerning joint distribution of the empirical density and of the empirical flows can be written in closed form for general Markov processes, including discrete-time Markov chains [96,[106][107][108][109][110], continuous-time Markov jump processes [57,106, and diffusion processes [57,70,109,110,114,115,127,130,131]. This explains why the Level 2.5 has been a major progress in the field of dynamical large deviations properties for Markov processes.…”

Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon

“…In these studies, as explained in detail in the two complementary papers [50,51] and in the HDR thesis [52], the Doob conditioned processes correspond to the processes that optimize the dynamical large deviations in the presence of the imposed constraints, showing the link with the field of stochastic control. It should be stressed that the corresponding rate functions at Level 2.5 are explicit for many Markov processes, including discrete-time Markov chains [72][73][74][75][76], continuous-time Markov jump processes [52,72, and Diffusion processes [52,65,75,76,80,81,93,96,97]. As incredible as it may seem, the very deep connections between the fields of Doob conditioning, of large deviations and of stochastic control are actually already present in the famous paper written in 1931 by E. Schrödinger [98], as discussed in detail in the recent detailed commentary [99] accompanying its english translation, as well as in the two reviews [100,101] written from the viewpoint of stochastic control and optimal transport.…”

“…Within the standard classification of the large deviation theory (see the reviews [115][116][117] and references therein), the Level 2 concerning the empirical density alone is usually not closed for non-equilibrium processes with steady currents, while the Level 3 concerning the whole empirical process is somewhat too general. By contrast, the Level 2.5 concerning the joint distribution of the empirical density and of the empirical flows can be written in closed form for general Markov processes, including discrete-time Markov chains [72][73][74][75][76]117], continuous-time Markov jump processes [52,72, and Diffusion processes [52,65,75,76,80,81,93,96,97]. This explains why the Level 2.5 can be considered as the cornerstone from which many other dynamical large deviations properties can be derived via the appropriate contractions.…”