Kiyoshi Kanazawa

Stochastic energetics for non-Gaussian processes.

By introducing a new stochastic integral, we investigate the energetics of classical stochastic systems driven by non-Gaussian white noises. In particular, we introduce a decomposition of the total energy difference into the work and the heat for each trajectory, and derive a formula to calculate the heat from experimental data on the dynamics. We apply our formulation and results to a Langevin system driven by a Poisson noise.

Minimal Model of Stochastic Athermal Systems: Origin of Non-Gaussian Noise

For a wide class of stochastic athermal systems, we derive Langevin-like equations driven by non-Gaussian noise, starting from master equations and developing a new asymptotic expansion. We found an explicit condition whereby the non-Gaussian properties of the athermal noise become dominant for tracer particles associated with both thermal and athermal environments. Furthermore, we derive an inverse formula to infer microscopic properties of the athermal bath from the statistics of the tracer particle. We apply our formulation to a granular motor under viscous friction and analytically obtain the angular velocity distribution function. Our theory demonstrates that the non-Gaussian Langevin equation is the minimal model of athermal systems.

Energetics of active fluctuations in living cells.

The nonequilibrium activity taking place in a living cell can be monitored with a tracer embedded in the medium. While microrheology experiments based on optical manipulation of such probes have become increasingly standard, we put forward a number of experiments with alternative protocols that, we claim, will provide insight into the energetics of active fluctuations. These are based on either performing thermodynamiclike cycles in control-parameter space or determining response to external perturbations of the confining trap beyond simple translation. We illustrate our proposals on an active itinerant Brownian oscillator modeling the dynamics of a probe embedded in a living medium.

Heat conduction induced by non-Gaussian athermal fluctuations.

We study the properties of heat conduction induced by non-Gaussian noises from athermal environments. We find that new terms should be added to the conventional Fourier law and the fluctuation theorem for the heat current, where its average and fluctuation are determined not only by the noise intensities but also by the non-Gaussian nature of the noises. Our results explicitly show the absence of the zeroth law of thermodynamics in athermal systems.

Asymptotic Derivation of Langevin-like Equation with Non-Gaussian Noise and Its Analytical Solution

We asymptotically derive a non-linear Langevin-like equation with non-Gaussian white noise for a wide class of stochastic systems associated with multiple stochastic environments, by developing the expansion method in our previous paper (Kanazawa et al. in Phys Rev Lett 114:090601–090606, 2015). We further obtain a full-order asymptotic formula of the steady distribution function in terms of a large friction coefficient for a non-Gaussian Langevin equation with an arbitrary non-linear frictional force. The first-order truncation of our formula leads to the independent-kick model and the higher-order correction terms directly correspond to the multiple-kicks effect during relaxation. We introduce a diagrammatic representation to illustrate the physical meaning of the high-order correction terms. As a demonstration, we apply our formula to a granular motor under Coulombic friction and get good agreement with our numerical simulations.

Granular rotor as a probe for a nonequilibrium bath.

This study numerically and analytically investigates the dynamics of a rotor under viscous or dry friction as a nonequilibrium probe of a granular gas. In order to demonstrate the role of the rotor as a probe for a nonequilibrium bath, the molecular dynamics (MD) simulation of the rotor is performed under viscous or dry friction surrounded by a steady granular gas under gravity. A one-to-one map between the velocity distribution function (VDF) of the granular gas and the angular distribution function for the rotor is theoretically derived. The MD simulation demonstrates that the one-to-one map accurately infers the local VDF of the granular gas from the angular VDF of the rotor, and vice versa.

Energy pumping in electrical circuits under avalanche noise.

We theoretically study energy pumping processes in an electrical circuit with avalanche diodes, where non-Gaussian athermal noise plays a crucial role. We show that a positive amount of energy (work) can be extracted by an external manipulation of the circuit in a cyclic way, even when the system is spatially symmetric. We discuss the properties of the energy pumping process for both quasistatic and finite-time cases, and analytically obtain formulas for the amounts of the work and the power. Our results demonstrate the significance of the non-Gaussianity in energetics of electrical circuits.

Markovian Stochastic Processes

We provide a brief and elementary review on the mathematical theory of stochastic processes. In particular, the correspondence between stochastic differential equations (SDEs) and master equations (MEs) is studied in detail. We first consider the Liouville equation as the simplest ME for an ordinary differential equation, and we next consider the MEs for SDEs driven by various Poisson noises. The Gaussian noise is introduced as a limit from the symmetric Poisson noise, and the Fokker-Planck equation is derived for an SDE driven by the Gaussian noise. We finally address an SDE driven by state-dependent Gaussian and Poisson noises, and studies the most general MEs in Markovian stochastic processes.

Analytical Solution to Nonlinear Non-Gaussian Langevin Equation

The non-Gaussian Langevin equation under nonlinear friction is derived from microscopic dynamics by generalizing the expansion in Chap. 7. We also present an analytical solution to non-Gaussian Langevin equation in the presence of nonlinear frictions; a full-order asymptotic formula for the steady PDF is derived for large frictional limit. The first-order approximation of our formula leads to the independent-kick model, and higher-order approximation corresponds to multiple kicks during relaxation. As a demonstration, a granular motor under dry friction is finally addressed.

Langevin Equation and Its Microscopic Derivation

The Langevin equation is reviewed as a fundamental equation for the Brownian motion. We first review the mathematical characters of the Langevin equation, and show its microscopic derivation by an asymptotic expansion of master equations (i.e., the system size expansion). By combining the kinetic theory, the Langevin equation is derived for various systems from microscopic dynamics. We finally discuss unsolved problems within the framework of the original system size expansion.