How a single number intertwines space and time in one dimensional diffusive systems

Pre-print : https://arxiv.org/abs/2206.08739

Title : Role of initial conditions in 1D diffusive systems : compressibility, hyperuniformity and long-term memory

Authors : Tirthankar Banerjee, Robert L. Jack and Michael E. Cates

At the starting point of equilibrium Statistical Mechanics lies the 'ergodic hypothesis' [1]. In simplified terms, a system is taken to be ergodic if the ensemble and time averages of relevant observables are mutually equal. The hypothesis holds as long as the observation time is sufficiently larger than any intrinsic timescale associated with the system's own dynamics. At such long timescales, ergodic systems end up in unique steady states, thus rendering the 'initial conditions' irrelevant.

Ergodicity naturally breaks down when there are diverging intrinsic (or microscopic) timescales in the system (for example, near phase transitions) [2,3]. In other words, 'non-ergodicity' is a signature of strong memory effects [4,5]. This lends an additional source of randomness to the problem as the system develops 'choices' of which steady state to end up in. Moreover, as there are diverging microscopic timescales, the observation time-scale is never really 'sufficiently larger' and therefore, the dependence on initial conditions can NOT be ruled out.

There is also a separate class of non-ergodic systems which lack any steady state [6]. For example, consider a simple 1D system: a gas of diffusive particles confined initially on the left-half plane [7,8]. Once we remove the confinement, the particles spread throughout the right-half plane, with the system never really reaching any steady state. This is an example of a transient system, where the observables themselves are 'dynamical quantities' (i.e., they explicitly depend on time). If the system is non-interacting, then it is logical to expect that the system will have an everlasting memory of its initial state. However, if one allows for complex interactions in diffusive systems, naively we would expect the system to forget about its initial state at long-enough times. Extraordinarily, extensive research on a wide class of interacting 1D diffusive systems has shown that the memory of initial conditions can persist forever [9,10]! 

To this end, people have considered two examples : (i) when the initial state is not allowed to fluctuate; such initial conditions are said to be 'quenched', and (ii) when the initial state is but the stationary state corresponding to the system's own dynamics; this state allows for fluctuations in each realization of the initial state, and the term 'annealed' is conveniently borrowed (from studies on disordered systems). It has been rigorously shown that under these two settings, the variances of dynamical quantities differ for a whole class of diffusive systems [9,10]. The variance in the 'quenched' case is less than the corresponding 'annealed' one (naively justifiable, as the annealed state has an additional source of entropy).

Although undoubtedly an intriguing result, the physics of such 1D diffusive systems is certainly not restricted to just these two choices of initial configurations. After all, there could be an initial profile which is not sampled from any equilibrium dynamics, let alone the system's own stationary state. The big picture confronts us with an obvious question : What is the general mechanism behind the manifestation of the long-lasting memory of initial conditions?

This is the question that we answer in the above article. We show that the information of the initial state can be entirely captured by a single number, α, which is related to the long-wavelength density fluctuations of the initial state. If the initial state is an equilibrium state (corresponding to the system's dynamics), then α is related to the compressibility of the system [11]. More generally, α is an asymptotic variance, and can be (technically) identified as the Fano factor [12]. What α captures is a simple fact: slow-relaxing modes of density fluctuations of the initial state are the sole source of long-lasting memory effects! This is what intertwines fluctuations in space to those in time.

Our result has remarkable consequences. First, initial states which correspond to α=0, are essentially 'hyperuniform' by definition [13]. All such initial states converge on the 'quenched' results of previous studies. This, therefore, allows us to contextualize the quenched situation alongside a universality class of hyperuniform initial states. Second, the 'annealed' setting from previous works turns out to be but one member of a continuous and infinite family of initial conditions, with each member parameterized by α only.

An important take-away point that emerges from our work is that the 'quenched'  setting does not have to conform to 'fixed' initial conditions. One can let the initial state fluctuate between infinite possibilities, but as long as each such state is hyperuniform on the 'macroscopic' scale, the variances of dynamical quantities, obtained by averaging over all such states, necessarily converge on the 'quenched' result(s).

Our analytical results were obtained for (i) current of non-interacting passive and active particles for step-like initial density profiles; and (ii) displacement of tracers in a class of strongly-interacting homogeneous systems.

It remains to be seen how large deviations (where higher order statistics of both the initial states and the dynamics conspire) affect the picture presented here. Also, there are interacting self-propelled systems where the memory of initial conditions is indeed washed out at finite times [14]. The underlying mechanism, is yet, elusive.

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