Exact distributions of currents and frenesy for Markov bridges

Thermodynamic cost for precision of general counting observables

We analytically derive universal bounds that describe the tradeoff between thermodynamic cost and precision in a sequence of events related to some internal changes of an otherwise hidden physical system. The precision is quantified by the fluctuations in either the number of events counted over time or the waiting times between successive events. Our results are valid for the same broad class of nonequilibrium driven systems considered by the thermodynamic uncertainty relation, but they extend to both time-symmetric and asymmetric observables. We show how optimal precision saturating the bounds can be achieved. For waiting-time fluctuations of asymmetric observables, a phase transition in the optimal configuration arises, where higher precision can be achieved by combining several signals.

Nonequilibrium diffusion processes via non-Hermitian electromagnetic quantum mechanics with application to the statistics of entropy production in the Brownian gyrator

The nonequilibrium Fokker-Planck dynamics in an arbitrary force field f[over ⃗](x[over ⃗]) in dimension N is revisited via the correspondence with the non-Hermitian quantum mechanics in a real scalar potential V(x[over ⃗]) and in a purely imaginary vector potential [-iA[over ⃗](x[over ⃗])] of real amplitude A[over ⃗](x[over ⃗]). The relevant parameters of irreversibility are then the N(N-1)/2 magnetic matrix elements B_{nm}(x[over ⃗])=-B_{mn}(x[over ⃗])=∂_{n}A_{m}(x[over ⃗])-∂_{m}A_{n}(x[over ⃗]), while it is enlightening to explore the corresponding gauge transformations of the vector potential A[over ⃗](x[over ⃗]). This quantum interpretation is even more fruitful to study the statistics of all the time-additive observables of the stochastic trajectories, since their generating functions correspond to the same quantum problem with additional scalar and/or vector potentials. Our main conclusion is that the analysis of their large deviations properties and the construction of the corresponding Doob conditioned processes can be drastically simplified via the choice of an appropriate gauge for each purpose. This general framework is then applied to the special time-additive observables of Ornstein-Uhlenbeck trajectories in dimension N, whose generating functions correspond to quantum propagators involving quadratic scalar potentials and linear vector potentials, i.e., to quantum harmonic oscillators in constant magnetic matrices. As simple illustrative example, we finally focus on the Brownian gyrator in dimension N=2 to compute the large deviations properties of the entropy production of its stochastic trajectories and to construct the corresponding conditioned processes having a given value of the entropy production per unit time.

Statistical physics of inhomogeneous transport: Unification of diffusion laws and inference from first-passage statistics

Characterization of composite materials, whose properties vary in space over microscopic scales, has become a problem of broad interdisciplinary interest. In particular, estimation of the inhomogeneous transport coefficients, e.g., the diffusion coefficient or the heat conductivity, which shape important processes in biology and engineering, is a challenging task. The analysis of such systems is further complicated because two alternative formulations of the inhomogeneous transport equations exist in the literature-the Smoluchowski and Fokker-Planck equations, which are also related to the so-called Ito-Stratonovich dilemma. Using the theory of statistical physics, we show that the two formulations, usually regarded as distinct models, are physically equivalent. From this result we develop efficient estimates for the transverse space-dependent diffusion coefficient in fluids near a phase boundary. Our method requires only measurements of escape probabilities and mean exit times of molecules leaving a narrow spatial region. We test our estimates in three case studies: (i) a Langevin model of a Büttikker-Landauer ratchet; atomistic molecular-dynamics simulations of liquid-water molecules in contact with (ii) vapor, and (iii) soap (surfactant) film which has promising applications in physical chemistry. Our analysis reveals that near the surfactant monolayer the mobility of water molecules is slowed down almost twice with respect to the bulk liquid. Moreover, the diffusion coefficient of water correlates with the transition from hydrophilic to hydrophobic parts of the film.

Cycle counts and affinities in stochastic models of nonequilibrium systems

For nonequilibrium systems described by finite Markov processes, we consider the number of times that a system traverses a cyclic sequence of states (a cycle). The joint distribution of the number of forward and backward instances of any given cycle is described by universal formulas which depend on the cycle affinity, but are otherwise independent of system details. We discuss the similarities and differences of this result to fluctuation theorems, and generalize the result to families of cycles, relevant under coarse graining. Finally, we describe the application of large deviation theory to this cycle-counting problem.

Statistics of Reaction Flux and Dynamical Activity Associated with a Diffusion-Influenced Ligand-Binding Reaction

In this paper, we consider a macromolecule with two competitive binding sites where a ligand can bind to and gives rise to a unicyclic reaction network consisting of four states-(i) a single state with both binding sites vacant, (ii) two states with one bound site and one free binding site, and (iii) an another single state with both sites occupied. We obtain probability densities of the time-integrated current along the clockwise direction and the dynamical activity or mean number of jumps between different states for finite times at a fast diffusion limit. On the other hand, in the diffusion-limited case, ligand diffusion between the two binding sites directly connects the mono-ligated states-changing the reaction scheme. Addition of the new reaction channel alters the precision of ligand occupancy to a single site, the mean dynamical activity, and the mean entropy production rate. All of these quantities are calculated with varying degrees of competition between the two sites for ligands, and we find that increase in the competition between the two sites decreases all above-mentioned additive functionals. The upper bound of precision associated with a single-site ligand occupancy for a diffusion-influenced reaction network is set by the mean dynamical activity (mean entropy production rate) at small (large) ligand concentrations at the steady-state limit.

Fluctuation-response inequality out of equilibrium

We present an approach to response around arbitrary out-of-equilibrium states in the form of a fluctuation-response inequality (FRI). We study the response of an observable to a perturbation of the underlying stochastic dynamics. We find that the magnitude of the response is bounded from above by the fluctuations of the observable in the unperturbed system and the Kullback-Leibler divergence between the probability densities describing the perturbed and the unperturbed system. This establishes a connection between linear response and concepts of information theory. We show that in many physical situations, the relative entropy may be expressed in terms of physical observables. As a direct consequence of this FRI, we show that for steady-state particle transport, the differential mobility is bounded by the diffusivity. For a "virtual" perturbation proportional to the local mean velocity, we recover the thermodynamic uncertainty relation (TUR) for steady-state transport processes. Finally, we use the FRI to derive a generalization of the uncertainty relation to arbitrary dynamics, which involves higher-order cumulants of the observable. We provide an explicit example, in which the TUR is violated but its generalization is satisfied with equality.