Nonequilibrium Probability Research Articles

Nonequilibrium probability currents in optically-driven colloidal suspensions

In the absence of directional motion it is often hard to recognize athermal fluctuations. Probability currents provide such a measure in terms of the rate at which they enclose area in the reduced phase space. We measure this area enclosing rate for trapped colloidal particles, where only one particle is driven. By combining experiment, theory, and simulation, we single out the effect of the different time scales in the system on the measured probability currents. In this controlled experimental setup, particles interact hydrodynamically. These interactions lead to a strong spatial dependence of the probability currents and to a local influence of athermal agitation. In a multiple-particle system, we show that even when the driving acts only on one particle, probability currents occur between other, non-driven particles. This may have significant implications for the interpretation of fluctuations in biological systems containing elastic networks in addition to a suspending fluid.

Probability flow solution of the Fokker–Planck equation

The method of choice for integrating the time-dependent Fokker–Planck equation (FPE) in high-dimension is to generate samples from the solution via integration of the associated stochastic differential equation (SDE). Here, we study an alternative scheme based on integrating an ordinary differential equation that describes the flow of probability. Acting as a transport map, this equation deterministically pushes samples from the initial density onto samples from the solution at any later time. Unlike integration of the stochastic dynamics, the method has the advantage of giving direct access to quantities that are challenging to estimate from trajectories alone, such as the probability current, the density itself, and its entropy. The probability flow equation depends on the gradient of the logarithm of the solution (its ‘score’), and so is a-priori unknown. To resolve this dependence, we model the score with a deep neural network that is learned on-the-fly by propagating a set of samples according to the instantaneous probability current. We show theoretically that the proposed approach controls the Kullback–Leibler (KL) divergence from the learned solution to the target, while learning on external samples from the SDE does not control either direction of the KL divergence. Empirically, we consider several high-dimensional FPEs from the physics of interacting particle systems. We find that the method accurately matches analytical solutions when they are available as well as moments computed via Monte-Carlo when they are not. Moreover, the method offers compelling predictions for the global entropy production rate that out-perform those obtained from learning on stochastic trajectories, and can effectively capture non-equilibrium steady-state probability currents over long time intervals.

Nonequilibrium currents in stochastic field theories: A geometric insight.

We introduce a formalism to study nonequilibrium steady-state probability currents in stochastic field theories. We show that generalizing the exterior derivative to functional spaces allows identification of the subspaces in which the system undergoes local rotations. In turn, this allows prediction of the counterparts in the real, physical space of these abstract probability currents. The results are presented for the case of the Active Model B undergoing motility-induced phase separation, which is known to be out of equilibrium but whose steady-state currents have not yet been observed, as well as for the Kardar-Parisi-Zhang equation. We locate and measure these currents and show that they manifest in real space as propagating modes localized in regions with nonvanishing gradients of the fields.

Activity-assisted barrier crossing of self-propelled colloids over parallel microgrooves.

We report a systematic study of the dynamics of self-propelled particles (SPPs) over a one-dimensional periodic potential landscape U_{0}(x), which is fabricated on a microgroove-patterned polydimethylsiloxane (PDMS) substrate. From the measured nonequilibrium probability density function P(x;F_{0}) of the SPPs, we find that the escape dynamics of the slow rotating SPPs across the potential landscape can be described by an effective potential U_{eff}(x;F_{0}), once the self-propulsion force F_{0} is included into the potential under the fixed angle approximation. This work demonstrates that the parallel microgrooves provide a versatile platform for a quantitative understanding of the interplay among the self-propulsion force F_{0}, spatial confinement by U_{0}(x), and thermal noise, as well as its effects on activity-assisted escape dynamics and transport of the SPPs.

The connection between dissipative chaos and nonequilibrium

Strong nonequilibrium conditions often give rise to dissipative chaos. However, the underlying connection between dissipative chaos and nonequilibrium remains unclear. Herein, we study this connection from the framework of the nonequilibrium potential-flux landscape. The nonequilibrium system dynamics are determined in part by the potential landscape associated with the steady-state probability distribution. The other part of the driving force for nonequilibrium dynamics comes from the steady-state probability flow, leading to the breaking of detailed balance. We find that dissipative chaos is driven mainly by the nonequilibrium probability flow. Using the Lorenz strange attractor model, we quantitatively demonstrate the close connection between the kinematic measures of the appearance of dissipative chaos and the degree of nonequilibrium. This work reveals the dynamic and thermodynamic origins of dissipative chaos beyond the interpretation based on nonlinearity. This study also provides a new understanding of the phase transitions in nonequilibrium systems.

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.

Entropy in Biochemical Failure

The ab-initio determination of the thermodynamic properties of the hydrolysis of the GTP gamma-phosphate in normal and abnormal cell functions of the RAS protein mutant Thr (Q61) leads to a description of energy cycle deviations in the abnormal mitogen-activated protein kinase cascade. 1 A predictive non-equilibrium probability statement describing the nonlinear changes for these open and finite-lifetime systems follows from reasonable enthalpy and entropy values between the normal and mutated forms based on structures of the GTPase states at the allosteric site. Recent advances in understanding entropy in terms of asymmetric and highly entropic catalysis lead to an investigation of the GTPase entropy, specifically with regard to a failure in catalysis of the phosphate fragment by a water hydrogen positioned by the enzyme. 2 Utilizing a simple atomic metal catalyst surface displacement model, a paradigm that reduces noise from the quantum entanglement plus atomic displacement terms results in the process entropy. The evaluation of entropies within the mixed ionic, covalent and entangled system requires a nonlinear Markovian approach utilizing von Neumann entropies achieved by a systematic accumulation of entangled potentials in a step-wise method. 3 , 4 Determination of the Hamiltonian for the entangled atomic state includes pure and mixed quantum states solved within the Araki–Leib triangle boundary resulting in only hard-entangled states, and the entanglement of Coulombic and Laughlin-like states can be evaluated by slicing the Hilbert spaces and solving the pure states, or mixed states separately, and then summing them. 5 Incorporating the resulting entanglement potentials as well as the Coulombic atomic displacement states into a derivative of the Fokker–Planck equation results in generated and produced entropy. 6

Fundamental study on the thermodynamic non-equilibrium freezing probability prediction model taking into account the super cooling phenomena in porous material

It is commonly considered that frost damage is caused by sudden freezing of supercooled water, which is a random phenomenon. Therefore, the aims of this study are to establish a prediction model for the probability of freezing until any lowest reached temperature, and to obtain the probability distribution function of the freezing point for the proposed analytical prediction model. First, theoretical prediction model for the probability of the instantaneous increment of ice content when lowest achieving temperature was known was derived based on these assumptions that building structure is an aggregation of small elements. Next, the freezing point measurement was carried out by using saturated mortar samples as the small element. As the results, it could be found that the first freezing due to supercooling occurred from -4 to -11 deg. C and the maximum probability was appeared at -7.5 deg. C. The average increment of ice content at every temperature closed to the 40 % volume of pore water until the thermodynamically-based freezing point. Moreover, the proposed method that can calculate the probability distributions of the instantaneous increment of ice content for any lowest achieving temperature from pore size distribution had good agreements with the measurement results.

Energy harvesting from anisotropic fluctuations.

We consider a rudimentary model for a heat engine, known as the Brownian gyrator, that consists of an overdamped system with two degrees of freedom in an anisotropic temperature field. Whereas the hallmark of the gyrator is a nonequilibrium steady-state curl-carrying probability current that can generate torque, we explore the coupling of this natural gyrating motion with a periodic actuation potential for the purpose of extracting work. We show that path lengths traversed in the manifold of thermodynamic states, measured in a suitable Riemannian metric, represent dissipative losses, while area integrals of a work density quantify work being extracted. Thus, the maximal amount of work that can be extracted relates to an isoperimetric problem, trading off area against length of an encircling path. We derive an isoperimetric inequality that provides a universal bound on the efficiency of all cyclic operating protocols, and a bound on how fast a closed path can be traversed before it becomes impossible to extract positive work. The analysis presented provides guiding principles for building autonomous engines that extract work from anisotropic fluctuations.

Emergent probability fluxes in confined microbial navigation

When the motion of a motile cell is observed closely, it appears erratic, and yet the combination of nonequilibrium forces and surfaces can produce striking examples of organization in microbial systems. While most of our current understanding is based on bulk systems or idealized geometries, it remains elusive how and at which length scale self-organization emerges in complex geometries. Here, using experiments and analytical and numerical calculations, we study the motion of motile cells under controlled microfluidic conditions and demonstrate that probability flux loops organize active motion, even at the level of a single cell exploring an isolated compartment of nontrivial geometry. By accounting for the interplay of activity and interfacial forces, we find that the boundary's curvature determines the nonequilibrium probability fluxes of the motion. We theoretically predict a universal relation between fluxes and global geometric properties that is directly confirmed by experiments. Our findings open the possibility to decipher the most probable trajectories of motile cells and may enable the design of geometries guiding their time-averaged motion.

Entropy production in model colloidal suspensions under shear via the fluctuation theorem.

Dissipative systems often exhibit novel and unexpected properties. This is, for instance, the case of simple liquids, which, when subjected to shear and after reaching a steady state, can exhibit a negative entropy production over finite length scales and timescales. This result, among others, is captured by nonequilibrium relations known as fluctuation theorems. Using nonequilibrium molecular dynamics simulations, we examine how, by fine-tuning the properties of the components of a complex fluid, we can steer the nonequilibrium response of the fluid. More specifically, we show how we control the nonequilibrium probability distribution for the shear stress and, in turn, how often states with a negative entropy production can occur. To achieve this, we start by characterizing how the size for the liquid matrix impacts the probability of observing negative entropy states, as well as the timescale over which these can be observed. We then measure how the addition of larger particles to this liquid matrix, i.e., simulating a model colloidal suspension, results in an increase in the occurrence of such states. This suggests how modifications in the composition of the mixture and in the properties of its components lead to an increase in the probability of observing states of negative entropy production and, thus, for the system to run in reverse.

The dynamic and thermodynamic origin of dissipative chaos: chemical Lorenz system

Chaos appears widely in various chemical and physical systems and is often accompanied by nonequilibrium due to its dissipative nature. However, it is still not clear how dissipative chaos is influenced by nonequilibrium conditions. Here, we study chaos from the perspective of nonequilibrium dynamics by considering a chemical Lorenz system. We found that its nonequilibrium nature can be quantified from the steady-state probability flux in the state space. The dynamic origin for the onset and offset of dissipative chaos was from the sudden appearance and disappearance of such nonequilibrium fluxes. Meanwhile, the dissipation associated with the flux as quantified by the entropy production rate provides the thermodynamic origin of dissipative chaos. Sharp changes in the degree of nonequilibrium also provide alternative quantitative indicators for the onset and offset of dissipative chaos.

Extension of generalized Kramers theory in the spatial diffusion regime: Diffusion of Brownian particles in periodic potential

Extension of generalized Kramers theory in the spatial diffusion regime to extensive noise parameters is illustrated through the internal Ornstein–Uhlenbeck noise-induced diffusion of a particle in one-dimensional periodic potential. In the current over population method, the Fokker–Planck equation does not have a non-equilibrium stationary-state solution for a large scope of Ornstein–Uhlenbeck noise parameters. Instead, an effective non-equilibrium stationary-state transition probability density can be obtained and can serve as the long-term mean transition probability density for the Langevin simulation of the escape rate. The analytical results of the diffusion coefficient in the spatial diffusion regime are in good agreement with the Langevin simulation results at a low temperature. This theoretical approach can be applied to Gaussian noise processes in potential barrier crossing problems, including the processes in the presence of external noises.

Nonequilibrium probability flux of a thermally driven micromachine.

We discuss the nonequilibrium statistical mechanics of a thermally driven micromachine consisting of three spheres and two harmonic springs [Y. Hosaka etal., J. Phys. Soc. Jpn. 86, 113801 (2017)JUPSAU0031-901510.7566/JPSJ.86.113801]. We obtain the nonequilibrium steady state probability distribution function of such a micromachine and calculate its probability flux in the corresponding configuration space. The resulting probability flux can be expressed in terms of a frequency matrix that is used to distinguish between a nonequilibrium steady state and a thermal equilibrium state satisfying detailed balance. The frequency matrix is shown to be proportional to the temperature difference between the spheres. We obtain a linear relation between the eigenvalue of the frequency matrix and the average velocity of a thermally driven micromachine that can undergo a directed motion in a viscous fluid. This relation is consistent with the scallop theorem for a deterministic three-sphere microswimmer.

Stochastic thermodynamics of nonharmonic oscillators in high vacuum.

We perform an analytic study on the stochastic thermodynamics of a small classical particle trapped in a time-dependent single-well potential in the highly underdamped limit. It is shown that the nonequilibrium probability density function for the system's energy is a Maxwell-Boltzmann distribution (as in equilibrium) with a closed form time-dependent effective temperature and fractional degrees of freedom. We also find that the solvable model satisfies the Crooks fluctuation theorem, as expected. Moreover, we compute the average work in this isothermal process and characterize analytically the optimal protocol for minimum work. The optimal protocol presents an initial and a final jump which correspond to adiabatic processes linked by a smooth exponential time-dependent part for all kinds of single-well potentials. Furthermore, we argue that this result connects two distinct relevant experimental setups for trapped nanoparticles, the levitated particle in a harmonic trap, and the free particle in a box, as they are limiting cases of the general single-well potential and display the time-dependent optimal protocols. Finally, we highlight the connection between our system and an equivalent model of a gas of Brownian particles.

Steady-state entropy: A proposal based on thermodynamic integration

Defining an entropy function out of equilibrium is an outstanding challenge. For stochastic lattice models in spatially uniform nonequilibrium steady states, definitions of temperature T and chemical potential μ have been verified using coexistence with heat and particle reservoirs. For an appropriate choice of exchange rates, T and μ satisfy the zeroth law, marking an important step in the development of steady-state thermodynamics. These results suggest that an associated steady-state entropy S_{th} be constructed via thermodynamic integration, using relations such as (∂S/∂E)_{V,N}=1/T, ensuring that derivatives of S_{th} with respect to energy and particle number yield the expected intensive parameters. We determine via direct calculation the stationary nonequilibrium probability distribution of the driven lattice gas with nearest-neighbor exclusion, the Katz-Lebowitz-Spohn driven lattice gas, and a two-temperature Ising model so that we may evaluate the Shannon entropy S_{S} as well as S_{th} defined above. Although the two entropies are identical in equilibrium (as expected), they differ out of equilibrium; for small values of the drive, D, we find |S_{S}-S_{th}|∝D^{2}, as expected on the basis of symmetry. We verify that S_{th} is not a state function: changes ΔS_{th} depend not only on the initial and final points, but also on the path in parameter space. The inequivalence of S_{S} and S_{th} implies that derivatives of S_{S} are not predictive of coexistence. In other words, a nonequilibrium steady state is not determined by maximizing the Shannon entropy. Our results cast doubt on the possibility of defining a state function that plays the role of a thermodynamic entropy for nonequilibrium steady states.

Dissipation, lag, and drift in driven fluctuating systems.

This work deals with thermostated fluctuating systems subjected to driven transformations of the internal energetics. The main focus is on generally multidimensional systems with continuous configurational degrees of freedom over which overdamped Markovian fluctuations take place (diffusive regime of the motion). Mutual bounds are established between the average energy dissipation, the deviation between nonequilibrium probability density and underlying equilibrium distribution due to the system's lag, and the statistical properties of the components of the directed flow induced by the transformation itself. The directed flow is here expressed in terms of time-dependent "drift velocity" associated with the probability current in a advection-like formulation of the nonstationary Fokker-Planck equation. Consideration of the drift makes that the bounds achieved here extend the inequality derived by Vaikuntanathan and Jarzynski [Europhys. Lett. 87, 60005 (2009)EULEEJ0295-507510.1209/0295-5075/87/60005] involving only dissipation and lag. The key relations are then specified for the so-called stochastic pumps, i.e., systems that reach a periodic steady state in response of cyclic transformations and that are prototypes of nonautonomous dissipative converters of input energy into directed motion; a one-dimensional case model is adopted to illustrate the main features. Complementary results concerning bounds between the evolution rates of dissipation and lag, valid for both overdamped and underdamped dynamics, are also presented.

Microscopic derivation of the hydrodynamics of active-Brownian-particle suspensions.

We derive the hydrodynamic equations of motion for a fluid of active particles described by underdamped Langevin equations that reduce to the active-Brownian-particle model, in the overdamped limit. The contraction into the hydrodynamic description is performed by locally averaging the particle dynamics with the nonequilibrium many-particle probability density, whose formal expression is found in the physically relevant limit of high friction through a multiple-time-scale analysis. This approach permits us to identify the conditions under which self-propulsion can be subsumed into the fluid stress tensor and thus to define systematically and unambiguously the local pressure of the active fluid.

Perturbative Large Deviation Analysis of Non-Equilibrium Dynamics

Macroscopic fluctuation theory has shown that a wide class of non-equilibrium stochastic dynamical systems obey a large deviation principle, but except for a few one-dimensional examples these large deviation principles are in general not known in closed form. We consider the problem of constructing successive approximations to an (unknown) large deviation functional and show that the non-equilibrium probability distribution the takes a Gibbs-Boltzmann form with a set of auxiliary (non-physical) energy functions. The expectation values of these auxiliary energy functions and their conjugate quantities satisfy a closed system of equations which can imply a considerable reduction of dimensionality of the dynamics. We show that the accuracy of the approximations can be tested self-consistently without solving the full non- equilibrium equations. We test the general procedure on the simple model problem of a relaxing 1D Ising chain.

Repeated Extraction of Work via Measurements on Nonequilibrium Two Particles Interacting in a Harmonic Trap

As an application of the generic theory for repeated extraction of work recently developed by the author [Phys. Rev. E 88, 032144 (2013)], we analyze interacting two particles in a harmonic trap subject to a heat bath. The observer changes the center of the trap as a control parameter according to the measurement outcome. The expressions for the time development of the nonequilibrium probability distribution function (PDF) of the particles and the control parameter is explicitly provided. It is demonstrated that the observer can extract work on the way to the stationary state, even when the successful extraction of work is not attained in the stationary state. The exact solution for the stationary PDF is provided, for which the relative-entropy work relation is confirmed to hold. The largest extracted work per unit time is also examined by incorporating an energy cost for each measurement. It is numerically demonstrated that a critical cost exists above which the successful stationary extraction of work i...