Breaking Of Detailed Balance Research Articles

Quantifying nonequilibrium dynamics and thermodynamics of cell fate decision making in yeast under pheromone induction.

Cellular responses to pheromone in yeast can range from gene expression to morphological and physiological changes. While signaling pathways are well studied, the cell fate decision-making during cellular polar growth is still unclear. Quantifying these cellular behaviors and revealing the underlying physical mechanism remain a significant challenge. Here, we employed a hidden Markov chain model to quantify the dynamics of cellular morphological systems based on our experimentally observed time series. The resulting statistics generated a stability landscape for state attractors. By quantifying rotational fluxes as the non-equilibrium driving force that tends to disrupt the current attractor state, the dynamical origin of non-equilibrium phase transition from four cell morphological fates to a single dominant fate was identified. We revealed that higher chemical voltage differences induced by a high dose of pheromone resulted in higher chemical currents, which will trigger a greater net input and, thus, more degrees of the detailed balance breaking. By quantifying the thermodynamic cost of maintaining morphological state stability, we demonstrated that the flux-related entropy production rate provides a thermodynamic origin for the phase transition in non-equilibrium morphologies. Furthermore, we confirmed that the time irreversibility in time series provides a practical way to predict the non-equilibrium phase transition.

Dissipation and energy propagation across scales in an active cytoskeletal material

Living systems are intrinsically nonequilibrium: They use metabolically derived chemical energy to power their emergent dynamics and self-organization. A crucial driver of these dynamics is the cellular cytoskeleton, a defining example of an active material where the energy injected by molecular motors cascades across length scales, allowing the material to break the constraints of thermodynamic equilibrium and display emergent nonequilibrium dynamics only possible due to the constant influx of energy. Notwithstanding recent experimental advances in the use of local probes to quantify entropy production and the breaking of detailed balance, little is known about the energetics of active materials or how energy propagates from the molecular to emergent length scales. Here, we use a recently developed picowatt calorimeter to experimentally measure the energetics of an active microtubule gel that displays emergent large-scale flows. We find that only approximately one-billionth of the system's total energy consumption contributes to these emergent flows. We develop a chemical kinetics model that quantitatively captures how the system's total thermal dissipation varies with ATP and microtubule concentrations but that breaks down at high motor concentration, signaling an interference between motors. Finally, we estimate how energy losses accumulate across scales. Taken together, these results highlight energetic efficiency as a key consideration for the engineering of active materials and are a powerful step toward developing a nonequilibrium thermodynamics of living systems.

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.

Emergence of local irreversibility in complex interacting systems.

Living systems are fundamentally irreversible, breaking detailed balance and establishing an arrow of time. But how does the evident arrow of time for a whole system arise from the interactions among its multiple elements? We show that the local evidence for the arrow of time, which is the entropy production for thermodynamic systems, can be decomposed. First, it can be split into two components: an independent term reflecting the dynamics of individual elements and an interaction term driven by the dependencies among elements. Adapting tools from nonequilibrium physics, we further decompose the interaction term into contributions from pairs of elements, triplets, and higher-order terms. We illustrate our methods on models of cellular sensing and logical computations, as well as on patterns of neural activity in the retina as it responds to visual inputs. We find that neural activity can define the arrow of time even when the visual inputs do not, and that the dominant contribution to this breaking of detailed balance comes from interactions among pairs of neurons.

The INSIDEOUT framework provides precise signatures of the balance of intrinsic and extrinsic dynamics in brain states

Finding precise signatures of different brain states is a central, unsolved question in neuroscience. We reformulated the problem to quantify the ‘inside out’ balance of intrinsic and extrinsic brain dynamics in brain states. The difference in brain state can be described as differences in the detailed causal interactions found in the underlying intrinsic brain dynamics. We used a thermodynamics framework to quantify the breaking of the detailed balance captured by the level of asymmetry in temporal processing, i.e. the arrow of time. Specifically, the temporal asymmetry was computed by the time-shifted correlation matrices for the forward and reversed time series, reflecting the level of non-reversibility/non-equilibrium. We found precise, distinguishing signatures in terms of the reversibility and hierarchy of large-scale dynamics in three radically different brain states (awake, deep sleep and anaesthesia) in electrocorticography data from non-human primates. Significantly lower levels of reversibility were found in deep sleep and anaesthesia compared to wakefulness. Non-wakeful states also showed a flatter hierarchy, reflecting the diversity of the reversibility across the brain. Overall, this provides signatures of the breaking of detailed balance in different brain states, perhaps reflecting levels of conscious awareness.

The onset of dissipative chaos driven by nonequilibrium conditions.

Dissipative chaos appears widely in various nonequilibrium systems; however, it is not clear how dissipative chaos originates from nonequilibrium. We discuss a framework based on the potential-flux approach to study chaos from the perspective of nonequilibrium dynamics. In this framework, chaotic systems possess a wide basinon the potential landscape, in which the rotational flux dominates the system dynamics, and chaos occurs with the appearance of this basin. In contrast, the probability flux is particularly associated with the detailed balance-breaking in nonequilibrium systems. This implies that the appearance of dissipative chaos is driven by nonequilibrium conditions.

Steady states of active Brownian particles interacting with boundaries

An active Brownian particle is a minimal model for a self-propelled colloid in a dissipative environment. Experiments and simulations show that, in the presence of boundaries and obstacles, active Brownian particle systems approach nontrivial nonequilibrium steady states with intriguing phenomenology, such as accumulation at boundaries, ratchet effects, and long-range depletion interactions. Nevertheless, theoretical analysis of these phenomena has proven difficult. Here, we address this theoretical challenge in the context of non-interacting particles in two dimensions, basing our analysis on the steady-state Smoluchowski equation for the one-particle distribution function. Our primary result is an approximation strategy that connects asymptotic solutions of the Smoluchowski equation to boundary conditions. We test this approximation against the exact analytic solution in a 2D planar geometry, as well as numerical solutions in circular and elliptic geometries. We find good agreement so long as the boundary conditions do not vary too rapidly with respect to the persistence length of particle trajectories. Our results are relevant for characterizing long-range flows and depletion interactions in such systems. In particular, our framework shows how such behaviors are connected to the breaking of detailed balance at the boundaries.

Landscape-Flux Framework for Nonequilibrium Dynamics and Thermodynamics of Open Hamiltonian Systems Coupled to Multiple Heat Baths.

We establish a nonequilibrium dynamic and thermodynamic formalism in the landscape-flux framework for open Hamiltonian systems in contact with multiple heat baths governed by stochastic dynamics. To systematically characterize nonequilibrium steady states, the nonequilibrium trinity construct is developed, which consists of detailed balance breaking, nonequilibrium potential landscape, and irreversible probability flux. We demonstrate that the temperature difference of the heat baths is the physical origin of detailed balance breaking, which generates the nonequilibrium potential landscape characterizing the nonequilibrium statistics and creates the irreversible probability flux signifying time irreversibility, with the latter two aspects closely connected. It is shown that the stochastic dynamics of the system can be formulated in the landscape-flux form, where the reversible force drives the conservative Hamiltonian dynamics, the irreversible force consisting of a landscape gradient force and an irreversible flux force drives the dissipative dynamics, and the stochastic force adds random fluctuations to the dynamics. The possible connection of the nonequilibrium trinity construct to nonequilibrium phase transitions is also suggested. A set of nonequilibrium thermodynamic equations, applicable to both nonequilibrium steady states and transient relaxation processes, is constructed. We find that an additional thermodynamic quantity, named the mixing entropy production rate, enters the nonequilibrium thermodynamic equations. It arises from the interplay between detailed balance breaking and transient relaxation, and it also relies on the conservative dynamics. At the nonequilibrium steady state, the heat flow, entropy flow, and entropy production are demonstrated to be thermodynamic manifestations of the nonequilibrium trinity construct. The general nonequilibrium formalism is applied to a class of solvable systems consisting of coupled harmonic oscillators. A more specific example of two harmonic oscillators coupled to two heat baths is worked out in detail. The example may facilitate connection with experiments.

Role of current fluctuations in nonreversible samplers.

It is known that the distribution of nonreversible Markov processes breaking the detailed balance condition converges faster to the stationary distribution compared to reversible processes having the same stationary distribution. This is used in practice to accelerate Markov chain Monte Carlo algorithms that sample the Gibbs distribution by adding nonreversible transitions or nongradient drift terms. The breaking of detailed balance also accelerates the convergence of empirical estimators to their ergodic expectation in the long-time limit. Here, we give a physical interpretation of this second form of acceleration in terms of currents associated with the fluctuations of empirical estimators using the level 2.5 of large deviations, which characterizes the likelihood of density and current fluctuations in Markov processes. Focusing on diffusion processes, we show that there is accelerated convergence because estimator fluctuations arise in general with current fluctuations, leading to an added large deviation cost compared to the reversible case, which shows no current. We study the current fluctuation most likely to arise in conjunction with a given estimator fluctuation and provide bounds on the acceleration, based on approximations of this current. We illustrate these results for the Ornstein-Uhlenbeck process in two dimensions and the Brownian motion on the circle.

Quantifying the potential and flux landscapes for nonequilibrium multiverse, a new scenario for time arrow

We propose a new scenario of nonequilibirum multiverse. We quantified the potential landscape and the flux landscape for the multiverse. The potential landscape quantifies the weight of each universe. When the terminal vacuum with zero (flat) or negative cosmological constant (AdS) have a chance to tunnel back to the normal universes with positive cosmological constant (dS) through the bounce suggested by the recent studies, the detailed balance of the populations of the multiverse can be broken. We found that the degree of the detailed balance breaking can be quantified by the underlying average flux and associated flux landscape, which gives arise to the dynamical origin of irreversibility and the time arrow of the multiverse. We also showed that the steady state of the multiverse is maintained by the thermodynamic cost quantified by the entropy production rate which is associated to the flux. This gives arise to thermodynamic origin of time irreversibility. On the other hand, we show that the evolution dynamics of the multiverse is determined by both the potential landscape and flux landscape. While the potential landscape determines the weight of the universes in the multiverse and attracts the multiverse to the steady state basins, the flux landscape provides the cycles or loops associating certain universes together. We show that terminal vacuum universes can have dominant weights or lowest potentials giving arise to a funnel shaped potential landscape, while terminal vacuum universes together with other normal universes including ours can form dominant cycles giving arise to a funnel shaped cycle flux landscape. This indicates that even our universe may not be distinct from others based on the probability measure, it may lie in the dominant cycle(s), leading to higher chance of being found. This may provide an additional way beyond the anthropic principle for identifying our universe.

Detailed Balance Broken by Catch Bond Kinetics Enables Mechanical-Adaptation in Active Materials.

Unlike nearly all engineered materials which contain bonds that weaken under load, biological materials contain "catch" bonds which are reinforced under load. Consequently, materials, such as the cell cytoskeleton, can adapt their mechanical properties in response to their state of internal, non-equilibrium (active) stress. However, how large-scale material properties vary with the distance from equilibrium is unknown, as are the relative roles of active stress and binding kinetics in establishing this distance. Through course-grained molecular dynamics simulations, the effect of breaking of detailed balance by catch bonds on the accumulation and dissipation of energy within a model of the actomyosin cytoskeleton is explored. It is found that the extent to which detailed balance is broken uniquely determines a large-scale fluid-solid transition with characteristic time-reversal symmetries. The transition depends critically on the strength of the catch bond, suggesting that active stress is necessary but insufficient to mount an adaptive mechanical response.

Generalized Fluctuation-Dissipation Theorem for Non-equilibrium Spatially Extended Systems

The fluctuation-dissipation theorem (FDT) connecting the response of the system to external perturbations with the fluctuations at thermodynamic equilibrium is a central result in statistical physics. There has been effort devoted to extending the FDT in several different directions since its original formulation. In this work we establish a generalized form of the FDT for spatially extended nonequilibrium stochastic systems described by continuous fields. The generalized FDT is formulated with the aid of the nonequilibrium force decomposition in the potential landscape and flux field theoretical framework. The general results are substantiated in the setting of the Ornstein-Uhlenbeck (OU) process and further illustrated by a more specific example worked out in detail. The key feature of this generalized FDT for nonequilibrium spatially extended systems is that it represents a ternary relation rather than a binary relation as the FDT for equilibrium systems does. In addition to the response function and the time derivative of the field-field correlation function that are present in the equilibrium FDT, the field-flux correlation function also enters the generalized FDT. This additional contribution originates from detailed balance breaking that signifies the nonequilibrium irreversible nature of the steady state. In the special case when the steady state is an equilibrium state obeying detailed balance, the field-flux correlation function vanishes and the ternary relation in the generalized FDT reduces to the binary relation in the equilibrium FDT.

Spectral characterization of non-Gaussian quantum noise: Keldysh approach and application to photon shot noise

Having accurate tools to describe non-classical, non-Gaussian environmental fluctuations is crucial for designing effective quantum control protocols and understanding the physics of underlying quantum dissipative environments. We show how the Keldysh approach to quantum noise characterization can be usefully employed to characterize frequency-dependent noise, focusing on the quantum bispectrum (i.e., frequency-resolved third cumulant). Using the paradigmatic example of photon shot noise fluctuations in a driven bosonic mode, we show that the quantum bispectrum can be a powerful tool for revealing distinctive non-classical noise properties, including an effective breaking of detailed balance by quantum fluctuations. The Keldysh-ordered quantum bispectrum can be directly accessed using existing noise spectroscopy protocols.

The Quantum Coherence Induced by Geometric Curvature of Gauge Field in Non-equilibrium Quantum Dynamics

The study of non-equilibrium quantum dynamics receives attentions recently. However, the nature and effects of nonequilibirumness such as detailed-balance breaking and the relationship to the underlying intrinsic geometry is still unclear. In this study, we uncover that a gauge field will be induced by the nonequilibriumness in the coherence representation and furthermore its internal geometrical curvature is directly related to the degree of detailed balance breaking. The nonequilibriumness of the quantum system induces an intrinsic geometric curvature which can enhance the quantum coherence, leading to the possibility of the space time origin for the non-local quantum correlations or the possibility of curved space time emergence from nonequilibrium quantum dynamics. We also uncovered that the internal curvature of the gauge field provides a bridge to connect the generalized quantum fluctuation dissipation theorem to the fluctuation theorem and time irreversibility of the quantum dynamics. The quantum time irreversibility is due to the path dependent factor along any particular path in an internal curved space, which is analogous to the Wilson lines (or Wilson loops) in Abelian gauge theory. We also found that the steady state quantum coherence disappears when the non-trivial internal curvature vanishes for the quantum system coupled with environments. When the curvature is relatively small indicating weak detailed balance breaking, the coherence increases as curvature increases. The internal curvature can provide a general and direct quantitative measure of the detail-balance breaking for any quantum/classical non-equilibrium systems, even without knowing the underlying steady state distribution or the steady state flux. Using an example of two harmonic oscillators coupled to two environments with different temperatures, we explicitly show the dependence of the internal curvature and quantum coherence.

A Catch Bond Suppresses Fluctuations in the Coordinated Actions of Myosin II Motors

We study the role of a catch bond, a bond that has an increased lifetime in response to an applied force, in the coordinated actions of myosin II motors. During muscle contraction, it is important for the actomyosin complex to have a steep response in the number of bound myosin motors with respect to an increasing tension, but also small fluctuations in the number of bound motors. This would not be possible at equilibrium, where response and fluctuations are linked through fluctuation-dissipation relationships. But, actomyosin operates out of equilibrium; the individual myosin motors undergo dissipative cycles (driven by ATP hydrolysis) of binding and unbinding. The key feature is that bond release, which has catch-bond behavior, is not the microscopic reverse of bond formation. This detailed-balance breaking, combined with the feedback control of the catch bond, induces a violation of fluctuation-dissipation to achieve a strong myosin II cooperativity that would not be possible at equilibrium but is needed for proper muscle contraction. We also show that a similar mechanism underlies the shear-induced binding of leukocytes and platelets to the blood vessel wall.

Scaling behavior of nonequilibrium measures in internally driven elastic assemblies.

Detecting and quantifying nonequilibrium activity is essential for studying internally driven assemblies, including synthetic active matter and complex living systems such as cells or tissue. We discuss a noninvasive approach of measuring nonequilibrium behavior based on the breaking of detailed balance. We focus on "cycling frequencies"-the average frequency with which the trajectories of pairs of degrees of freedom revolve in phase space-and explain their connection with other nonequilibrium measures, including the area enclosing rate and the entropy production rate. We test our approach on simple toy models composed of elastic networks immersed in a viscous fluid with site-dependent internal driving. We prove both numerically and analytically that the cycling frequencies obey a power law as a function of distance between the tracked degrees of freedom. Importantly, the behavior of the cycling frequencies contains information about the dimensionality of the system and the amplitude of active noise. The mapping we use in our analytical approach thus offers a convenient framework for predicting the behavior of two-point nonequilibrium measures for a given activity distribution in the network.

Stochastic Thermodynamics of Oscillators' Networks.

We apply the stochastic thermodynamics formalism to describe the dynamics of systems of complex Langevin and Fokker-Planck equations. We provide in particular a simple and general recipe to calculate thermodynamical currents, dissipated and propagating heat for networks of nonlinear oscillators. By using the Hodge decomposition of thermodynamical forces and fluxes, we derive a formula for entropy production that generalises the notion of non-potential forces and makes transparent the breaking of detailed balance and of time reversal symmetry for states arbitrarily far from equilibrium. Our formalism is then applied to describe the off-equilibrium thermodynamics of a few examples, notably a continuum ferromagnet, a network of classical spin-oscillators and the Frenkel-Kontorova model of nano friction.

Non-Markovian nonequilibrium information dynamics

We probe into the dynamics of interacting non-Markovian information systems. The stochastic dynamics of information has two aspects: the self-evolution and interaction. We show that self-evolution of a non-Markovian information system can be described by a Markov-type master equation with memory dependence. We also reveal that the interaction between systems can be fully embodied into the information dynamics of the composite information system. To characterize time irreversibility of the self-evolution and the interaction, we apply the landscape-flux theory to both stochastic and thermal information dynamics. The driving force of the nonequilibrium information dynamics can be decomposed into time-reversible (detailed balance preserving landscape part) and -irreversible (detailed balance breaking nonequilibrium flux part) parts. The time-irreversible part of the driving force fully depicts the time-irreversibility behavior in the stochastic dynamics. The time irreversibility of the interactions between systems reflected in nonequilibrium thermodynamics can be seen in the decomposition of the mutual information rate which corresponds to decomposition of the driving force. In particular, the time-irreversible part of mutual information rate reveals the underlying relationship among the entropy production rates of the information systems. We propose the finite memory approximation method and demonstrate that the above mentioned features can be found in a wide class of non-Markovian nonequilibrium information systems. Finally, we derive the lower and upper bounds for informational entities under concern with clear meanings.

Active Brownian particles driven by constant affinity

Experimental realizations of self-propelled colloidal Janus particles exploit the conversion of free energy into directed motion. One route is phoretic mechanisms that can be modeled schematically as the interconversion of two chemical species. Here we consider the situation when the difference of chemical potential between the two species (the driving affinity) can be assumed to be constant, and we derive the thermodynamically consistent equations of motion. In contrast to the standard model of active Brownian particles parametrized by a constant self-propulsion speed, this yields a non-constant speed that depends on the potential energy of the suspension. This approach allows to consistently model the breaking of detailed balance and the accompanying entropy production without non-conservative forces.

Potential landscape and flux field theory for turbulence and nonequilibrium fluid systems

Turbulence is a paradigm for far-from-equilibrium systems without time reversal symmetry. To capture the nonequilibrium irreversible nature of turbulence and investigate its implications, we develop a potential landscape and flux field theory for turbulent flow and more general nonequilibrium fluid systems governed by stochastic Navier–Stokes equations. We find that equilibrium fluid systems with time reversibility are characterized by a detailed balance constraint that quantifies the detailed balance condition. In nonequilibrium fluid systems with nonequilibrium steady states, detailed balance breaking leads directly to a pair of interconnected consequences, namely, the non-Gaussian potential landscape and the irreversible probability flux, forming a ‘nonequilibrium trinity’. The nonequilibrium trinity characterizes the nonequilibrium irreversible essence of fluid systems with intrinsic time irreversibility and is manifested in various aspects of these systems. The nonequilibrium stochastic dynamics of fluid systems including turbulence with detailed balance breaking is shown to be driven by both the non-Gaussian potential landscape gradient and the irreversible probability flux, together with the reversible convective force and the stochastic stirring force. We reveal an underlying connection of the energy flux essential for turbulence energy cascade to the irreversible probability flux and the non-Gaussian potential landscape generated by detailed balance breaking. Using the energy flux as a center of connection, we demonstrate that the four-fifths law in fully developed turbulence is a consequence and reflection of the nonequilibrium trinity. We also show how the nonequilibrium trinity can affect the scaling laws in turbulence.