Time evolution of entropy associated with diffusivity fluctuations: diffusing diffusivity approach

Abstract

It has experimentally been found by Lampo et al. [Biophys. J. 112, 532 (2017)] that, for two different types of cell, the distribution of the diffusivities of RNAprotein particles over cytoplasm obeys an exponential law. Then, an interesting issue has been pointed out: this exponential distribution is the maximal entropy distribution. Here, time evolution of entropy associated with local fluctuations of the diffusivity is studied.The entropy rate under the diffusing diffusivity equation, which admits the exponential fluctuation as its stationary solution, is shown to be positive. The present result is expected to be useful for studying the dynamics of diffusivity fluctuations. Furthermore, the distribution of time being required for characteristic displacement of the RNA-protein particle is found to decay as a power law. A comment is also made on a formal analogy with the thermodynamic relation concerning temperature.PACS number(s): 87.15. Ya, 87.15.Vv,

Superstatistical modelling of protein diffusion dynamics in bacteria

“…It may be worth pointing out that the power-law nature in equation (1.3) is non-trivial, since it is apparently different from the exponential law reported, for example, in [14] (see also [20][21][22] for an entropic approach to this law).…”

Fokker-Planck approach to non-Gaussian normal diffusion: Hierarchical dynamics for diffusing diffusivity

“…A description of such a probability distribution by use of the maximum entropy method is discussed in Ref. [16].…”

Fluctuating Diffusivity of RNA-Protein Particles: Analogy with Thermodynamics

“…The above idea has been further supported by explicitly showing [ 10 ] that the entropy production rate becomes manifestly positive under the mechanism of the so-called “diffusing diffusivity” [ 13 ], which offers a description of time evolution of diffusivity fluctuations and leads to the exponential fluctuation distribution as a stationary solution of its evolution equation (see, e.g., [ 14 ] for a recent development, where emergence of correlation time characterizing diffusivity dynamics has been discussed). As shown in [ 1 ], this mechanism combined with the approach of fractional Brownian motion [ 15 ] modeling the subdiffusion of the RNA-protein particles yields the non-Gaussian displacement distribution observed in the experiment.…”

“…However, the following idea has been suggested [ 1 ]: the distribution in Equation (2) is the maximal entropy distribution. To accomplish this, a maximum-entropy-principle approach has been developed in a recent work in [ 10 ]. Its basic observation is as follows.…”

“…Now, a comment, which motivates our present work, has also been made on an analogy with the thermodynamic relation concerning temperature [ 10 ]. Let us consider the thermodynamic-like situation in such a way that slowly changes, the time scale of which should be much larger than that of variation of diffusivity fluctuations mentioned above.…”