Entropy production and irreversibility of dissipative trajectories in electric circuits

Irreversible dynamics of a continuum driven by active matter

We study the fluctuational behavior of overdamped elastic filaments (e.g., strings or rods) driven by active matter which induces irreversibility. The statistics of discrete normal modes are translated into the continuum of the position representation which allows discernment of the spatial structure of dissipation and fluctuational work done by the active forces. The mapping of force statistics onto filament statistics leads to a generalized fluctuation-dissipation relation which predicts the components of the stochastic area tensor and its spatial proxy, the irreversibility field. We illustrate the general theory with explicit results for a tensioned string between two fixed endpoints. Plots of the stochastic area tensor components in the discrete plane of mode pairs reveal how the active forces induce spatial correlations of displacement along the filament. The irreversibility field provides additional quantitative insight into the relative spatial distributions of fluctuational work and dissipative response.

Coexistence of distinct nonuniform nonequilibrium steady states in Ehrenfest multiurn model on a ring

The recently proposed Ehrenfest M-urn model with interactions on a ring is considered as a paradigm model which can exhibit a variety of distinct nonequilibrium steady states. Unlike the previous three-urn model on a ring which consists of a uniform steady state and a nonuniform nonequilibrium steady state, it is found that for even M≥4, an additional nonequilibrium steady state can coexist with the original ones. Detailed analysis reveals that this additional nonequilibrium steady state emerged via a pitchfork bifurcation which cannot occur if M is odd. Properties of this nonequilibrium steady state, such as stability, and steady-state flux are derived analytically for the four-urn case. The full phase diagram with the phase boundaries is also derived explicitly. The associated thermodynamic stability is also analyzed, confirming its stability. These theoretical results are also explicitly verified by direct Monte Carlo simulations for the three-urn and four-urn ring models.

Stochastic line integrals and stream functions as metrics of irreversibility and heat transfer

Stochastic line integrals are presented as a useful metric for quantitatively characterizing irreversibility and detailed balance violation in noise-driven dynamical systems. A particular realization is the stochastic area, recently studied in coupled electrical circuits. Here we provide a general framework for understanding properties of stochastic line integrals and clarify their implementation for experiments and simulations. For two-dimensional systems, stochastic line integrals can be expressed in terms of a stream function, the sign of which determines the orientation of nonequilibrium steady-state probability currents. Theoretical results are supported by numerical studies of an overdamped two-dimensional mass-spring system driven out of equilibrium. Additionally, the stream function permits analytical understanding of the scaling dependence of stochastic area growth rate on key parameters such as the noise strength for both linear and nonlinear springs.

Generation of virtual potentials by controlled feedback in electric circuit systems

Electric circuits influenced by thermal noise are analogous to confined Brownian particles and can be an alternative and convenient scheme for studying stochastic thermodynamics. Here we experimentally demonstrate an effective technique of generating tunable potentials for Brownian dynamics in an electric circuit, realized by external controlled feedback. We present two illustrative examples of one-dimensional virtual potentials: static harmonic potential and time-varying double-well potential. The thermal noises of both cases undergo equivalent Brownian dynamics as if they were in the authentic potentials as long as the feedback is fast enough to respond to the designed potentials. The results show that the electric circuit provides a simple, effective, and programmable scheme to study the feedback-controlled virtual potential.

Infima statistics of entropy production in an underdamped Brownian motor

The second law of thermodynamics states that the entropy never decreases for isolated macroscopical systems, which defines the arrow of time. For small systems, although the entropy increases on average, due to strong fluctuation, it may encounter a temporary decrease. The probability of negative entropy production follows the fluctuation theorem. Recently, it has been demonstrated theoretically the infima law that there exists a lower bound for the average values of the minima of the negative entropy production, which is -k_{B}. In this paper, we have constructed a horizontal Brownian motor immersed in a granular gas, whose dynamics is governed by the underdamped stochastic process. By recording the angular motion of the motor and measuring the key parameters of the system, we experimentally demonstrate that, despite the nonideal elements in the experiments and that the complex underlying dynamics, the average value of the minima of the negative entropy production is still bounded by -k_{B}, which may invoke further theoretical investigations of the applicability of the infima law in nonideal realistic small systems.

Experimental metrics for detection of detailed balance violation

We report on the measurement of detailed balance violation in a coupled, noise-driven linear electronic circuit consisting of two nominally identical RC elements that are coupled via a variable capacitance. The state variables are the time-dependent voltages across each of the two primary capacitors, and the system is driven by independent noise sources in series with each of the resistances. From the recorded time histories of these two voltages, we quantify violations of detailed balance by three methods: (1) explicit construction of the probability current density, (2) constructing the time-dependent stochastic area, and (3) constructing statistical fluctuation loops. In comparing the three methods, we find that the stochastic area is relatively simple to implement and computationally inexpensive and provides a highly sensitive means for detecting violations of detailed balance.

Electrical autonomous Brownian gyrator

We study experimentally and theoretically the steady-state dynamics of a simple stochastic electronic system featuring two resistor-capacitor circuits coupled by a third capacitor. The resistors are subject to thermal noises at real temperatures. The voltage fluctuation across each resistor can be compared to a one-dimensional Brownian motion. However, the collective dynamical behavior, when the resistors are subject to distinct thermal baths, is identical to that of a Brownian gyrator, as first proposed by Filliger and Reimann [Phys. Rev. Lett. 99, 230602 (2007)PRLTAO0031-900710.1103/PhysRevLett.99.230602]. The average gyrating dynamics is originated from the absence of detailed balance due to unequal thermal baths. We look into the details of this stochastic gyrating dynamics, its dependences on the temperature difference and coupling strength, and the mechanism of heat transfer through this simple electronic circuit. Our work affirms the general principle and the possibility of a Brownian ratchet working near room temperature scale.

Colloidal dynamics over a tilted periodic potential: Forward and reverse transition probabilities and entropy production in a nonequilibrium steady state

We report a systematic study of the forward and reverse transition probability density functions (TPDFs) and entropy production in a nonequilibrium steady state (NESS). The NESS is realized in a two-layer colloidal system, in which the bottom-layer colloidal crystal provides a two-dimensional periodic potential U_{0}(x,y) for the top-layer diffusing particles. By tilting the sample at an angle with respect to gravity, a tangential component of the gravitational force F is applied to the diffusing particles, which breaks the detailed balance (DB) condition and generates a steady particle flux along the [1,0] crystalline orientation. While both the measured forward and reverse TPDFs reveal interesting space-time dependence, their ratio is found to be independent of time and obeys a DB-like relation. The experimental results are in good agreement with the theoretical predictions. This study thus provides a better understanding on how entropy is generated and heat is dissipated to the reservoir during a NESS transition process. It also demonstrates the applications of the two-layer colloidal system in the study of NESS transition dynamics.