Breaking Of Detailed Balance Research Articles

Funneled potential and flux landscapes dictate the stabilities of both the states and the flow: Fission yeast cell cycle.

Using fission yeast cell cycle as an example, we uncovered that the non-equilibrium network dynamics and global properties are determined by two essential features: the potential landscape and the flux landscape. These two landscapes can be quantified through the decomposition of the dynamics into the detailed balance preserving part and detailed balance breaking non-equilibrium part. While the funneled potential landscape is often crucial for the stability of the single attractor networks, we have uncovered that the funneled flux landscape is crucial for the emergence and maintenance of the stable limit cycle oscillation flow. This provides a new interpretation of the origin for the limit cycle oscillations: There are many cycles and loops existed flowing through the state space and forming the flux landscapes, each cycle with a probability flux going through the loop. The limit cycle emerges when a loop stands out and carries significantly more probability flux than other loops. We explore how robustness ratio (RR) as the gap or steepness versus averaged variations or roughness of the landscape, quantifying the degrees of the funneling of the underlying potential and flux landscapes. We state that these two landscapes complement each other with one crucial for stabilities of states on the cycle and the other crucial for the stability of the flow along the cycle. The flux is directly related to the speed of the cell cycle. This allows us to identify the key factors and structure elements of the networks in determining the stability, speed and robustness of the fission yeast cell cycle oscillations. We see that the non-equilibriumness characterized by the degree of detailed balance breaking from the energy pump quantified by the flux is the cause of the energy dissipation for initiating and sustaining the replications essential for the origin and evolution of life. Regulating the cell cycle speed is crucial for designing the prevention and curing strategy of cancer.

Nonequilibrium dynamics of probe filaments in actin-myosin networks.

Active dynamic processes of cells are largely driven by the cytoskeleton, a complex and adaptable semiflexible polymer network, motorized by mechanoenzymes. Small dimensions, confined geometries, and hierarchical structures make it challenging to probe dynamics and mechanical response of such networks. Embedded semiflexible probe polymers can serve as nonperturbing multiscale probes to detect force distributions in active polymer networks. We show here that motor-induced forces transmitted to the probe polymers are reflected in nonequilibrium bending dynamics, which we analyze in terms of spatial eigenmodes of an elastic beam under steady-state conditions. We demonstrate how these active forces induce correlations among the mode amplitudes, which furthermore break time-reversal symmetry. This leads to a breaking of detailed balance in this mode space. We derive analytical predictions for the magnitude of resulting probability currents in mode space in the white-noise limit of motor activity. We relate the structure of these currents to the spatial profile of motor-induced forces along the probe polymers and provide a general relation for observable currents on two-dimensional hyperplanes.

Landscape and flux theory of non-equilibrium open economy

The economy is open and never in true equilibrium due to the exchanges with outside. However, most of the quantitative studies have been focused on the equilibrium economy. Despite of the recent efforts, it is still challenging to formulate a quantitative theory for uncovering the principles of non-equilibrium open economy. In this study, we developed a landscape and flux theory for non-equilibrium economy. We quantified the states of economy and identify the multi-stable states as the basins of attractions on the underlying landscape. We found the global driving force of the non-equilibrium economy is determined by both the underlying landscape gradient and the curl probability flux measuring the degree of non-equilibriumness through the detailed balance breaking. The non-equilibrium thermodynamics, the global stability, the optimal path and speed of the non-equilibrium economy can be formulated and quantified. In the conventional economy, the supply and demand usually has only one equilibrium. By considering nonlinear supply–demand dynamics, we found that both bi-stable states and limit cycle oscillations can emerge. By shifting the slope of demand curve, we can see how the bi-stability transforms to the limit cycle dynamics and vice versa. By parallel shifting the demand curve, we can also see how the monopoly, the competition, and the bistable monopoly and competition states emerge and transform to one other. We can also see how the mono-stable monopoly, the limit cycle and the mono-stable competition states emerge and transform to one another.

Nonequilibrium-induced enhancement of dynamical quantum coherence and entanglement of spin arrays

The random magnetic field produced by nuclear spins has long been viewed as the dominating source of decoherence in the quantum-dot based spins. Here we obtain in both exact and analytical manner the dynamics of spin qubits coupled to nuclear spin environments via the hyperfine interaction, going beyond the weak system-bath interaction and Markovian approximation. We predict that the detailed-balance breaking produced by chemical potential gradient in nuclear baths leads to the rapid oscillations of populations, quantum coherence and entanglement, which are absent in the conventional case (i.e., Overhauser noise). This is attributed to the nonequilibrium feature of the system as shown in the relation between the oscillation period and the chemical potential imbalance. Our results reveal the essentiality of nonequilibriumness with detailed-balance breaking for enhancing the dynamical coherence and entanglement of spin qubits. Moreover, our exact solution explicitly demonstrates that the non-Markovian bath comprised by nuclear spins can preserve the collective quantum state, due to the recovery of coherence. Finally, we propose an experiment using ultracold trapped ions to observe these nonequilibrium and memory effects.

Quantification of motor network dynamics in Parkinson's disease by means of landscape and flux theory.

The basal ganglia neural circuit plays an important role in motor control. Despite the significant efforts, the understanding of the principles and underlying mechanisms of this modulatory circuit and the emergence of abnormal synchronized oscillations in movement disorders is still challenging. Dopamine loss has been proved to be responsible for Parkinson’s disease. We quantitatively described the dynamics of the basal ganglia-thalamo-cortical circuit in Parkinson’s disease in terms of the emergence of both abnormal firing rates and firing patterns in the circuit. We developed a potential landscape and flux framework for exploring the modulatory circuit. The driving force of the circuit can be decomposed into a gradient of the potential, which is associated with the steady-state probability distributions, and the curl probability flux term. We uncovered the underlying potential landscape as a Mexican hat-shape closed ring valley where abnormal oscillations emerge due to dopamine depletion. We quantified the global stability of the network through the topography of the landscape in terms of the barrier height, which is defined as the potential difference between the maximum potential inside the ring and the minimum potential along the ring. Both a higher barrier and a larger flux originated from detailed balance breaking result in more stable oscillations. Meanwhile, more energy is consumed to support the increasing flux. Global sensitivity analysis on the landscape topography and flux indicates how changes in underlying neural network regulatory wirings and external inputs influence the dynamics of the system. We validated two of the main hypotheses(direct inhibition hypothesis and output activation hypothesis) on the therapeutic mechanism of deep brain stimulation (DBS). We found GPe appears to be another effective stimulated target for DBS besides GPi and STN. Our approach provides a general way to quantitatively explore neural networks and may help for uncovering more efficacious therapies for movement disorders.

Driven to peak

A curious peak in the distribution describing stochastic switching in bacterial motility had researchers confounded. But a careful study performed under varying mechanical conditions has now revealed that the breaking of detailed balance is to blame.

Fluctuations in reactive networks subject to extrinsic noise studied in the framework of the chemical Langevin equation.

Theoretical and experimental studies have shown that the fluctuations of in vivo systems break the fluctuation-dissipation theorem. One can thus ask what information is contained in the correlation functions of protein concentrations and how they relate to the response of the reactive network to a perturbation. Answers to these questions are of prime importance to extract meaningful parameters from the in vivo fluorescence correlation spectroscopy data. In this paper we study the fluctuations of the concentration of a reactive species involved in a cyclic network that is in a nonequilibrium steady state perturbed by a noisy force, taking into account both the breaking of detailed balance and extrinsic noises. Using a generic model for the network and the extrinsic noise, we derive a chemical Langevin equation that describes the dynamics of the system, we determine the expressions of the correlation functions of the concentrations, and we estimate the deviation of the fluctuation-dissipation theorem and the range of parameters in which an effective temperature can be defined.

Fluctuation-dissipation theorem for nonequilibrium quantum systems

We present a fluctuation-dissipation theorem (FDT) for nonequilibrium quantum systems with detailed-balance breaking, which deviates from the conventional form of the FDT for equilibrium systems preserving detailed balance. Using the phase space formulation of quantum mechanics and the potential-flux landscape framework, we find that the response function of nonequilibrium quantum systems to external perturbations contains a nontrivial contribution from the quantum curl flux quantifying detailed-balance breaking in the steady state, in addition to the correlation function of observables in the steady state representing the contribution of spontaneous fluctuations which is also present in the equilibrium FDT. We illustrate our general formalism with a harmonic oscillator coupled to two heat baths, and show that the nonequilibrium FDT reduces to the conventional expression at the equilibrium condition when the flux contribution vanishes.

Physical mechanism of mind changes and tradeoffs among speed, accuracy, and energy cost in brain decision making: Landscape, flux, and path perspectives**Project supported by the National Natural Science Foundation of China (Grant Nos. 21190040, 91430217, and 11305176).

Cognitive behaviors are determined by underlying neural networks. Many brain functions, such as learning and memory, have been successfully described by attractor dynamics. For decision making in the brain, a quantitative description of global attractor landscapes has not yet been completely given. Here, we developed a theoretical framework to quantify the landscape associated with the steady state probability distributions and associated steady state curl flux, measuring the degree of non-equilibrium through the degree of detailed balance breaking for decision making. We quantified the decision-making processes with optimal paths from the undecided attractor states to the decided attractor states, which are identified as basins of attractions, on the landscape. Both landscape and flux determine the kinetic paths and speed. The kinetics and global stability of decision making are explored by quantifying the landscape topography through the barrier heights and the mean first passage time. Our theoretical predictions are in agreement with experimental observations: more errors occur under time pressure. We quantitatively explored two mechanisms of the speed-accuracy tradeoff with speed emphasis and further uncovered the tradeoffs among speed, accuracy, and energy cost. Our results imply that there is an optimal balance among speed, accuracy, and the energy cost in decision making. We uncovered the possible mechanisms of changes of mind and how mind changes improve performance in decision processes. Our landscape approach can help facilitate an understanding of the underlying physical mechanisms of cognitive processes and identify the key factors in the corresponding neural networks.

Matrix ansatz for the fluctuations of the current in the ASEP with open boundaries

The asymmetric simple exclusion process (ASEP) is one of the most extensively studied models in non-equilibrium statistical mechanics. The macroscopic particle current produced in its steady state is directly related to the breaking of detailed balance, and is therefore a physical quantity of particular interest. In this paper, we build a matrix product ansatz which allows us to access the exact statistics of the fluctuations of that current for finite sizes, as well as the probabilities of configurations conditioned on the mean current. We also show how this ansatz can be used for the periodic ASEP and how it translates into the language of the XXZ spin chain.

Energetic costs of cellular computation

Cells often perform computations in order to respond to environmental cues. A simple example is the classic problem, first considered by Berg and Purcell, of determining the concentration of a chemical ligand in the surrounding media. On general theoretical grounds, it is expected that such computations require cells to consume energy. In particular, Landauer's principle states that energy must be consumed in order to erase the memory of past observations. Here, we explicitly calculate the energetic cost of steady-state computation of ligand concentration for a simple two-component cellular network that implements a noisy version of the Berg-Purcell strategy. We show that learning about external concentrations necessitates the breaking of detailed balance and consumption of energy, with greater learning requiring more energy. Our calculations suggest that the energetic costs of cellular computation may be an important constraint on networks designed to function in resource poor environments, such as the spore germination networks of bacteria.

The potential and flux landscape theory of evolution

We established the potential and flux landscape theory for evolution. We found explicitly the conventional Wright's gradient adaptive landscape based on the mean fitness is inadequate to describe the general evolutionary dynamics. We show the intrinsic potential as being Lyapunov function(monotonically decreasing in time) does exist and can define the adaptive landscape for general evolution dynamics for studying global stability. The driving force determining the dynamics can be decomposed into gradient of potential landscape and curl probability flux. Non-zero flux causes detailed balance breaking and measures how far the evolution from equilibrium state. The gradient of intrinsic potential and curl flux are perpendicular to each other in zero fluctuation limit resembling electric and magnetic forces on electrons. We quantified intrinsic energy, entropy and free energy of evolution and constructed non-equilibrium thermodynamics. The intrinsic non-equilibrium free energy is a Lyapunov function. Both intrinsic potential and free energy can be used to quantify the global stability and robustness of evolution. We investigated an example of three allele evolutionary dynamics with frequency dependent selection (detailed balance broken). We uncovered the underlying single, triple, and limit cycle attractor landscapes. We found quantitative criterions for stability through landscape topography. We also quantified evolution pathways and found paths do not follow potential gradient and are irreversible due to non-zero flux. We generalized the original Fisher's fundamental theorem to the general (i.e., frequency dependent selection) regime of evolution by linking the adaptive rate with not only genetic variance related to the potential but also the flux. We show there is an optimum potential where curl flux resulting from biotic interactions of individuals within a species or between species can sustain an endless evolution even if the physical environment is unchanged. We offer a theoretical basis for explaining the corresponding Red Queen hypothesis proposed by Van Valen. Our work provides a theoretical foundation for evolutionary dynamics.

Addendum to “Event-chain Monte Carlo algorithms for hard-sphere systems”

We extend the event-chain Monte Carlo algorithm from hard-sphere interactions to general potentials. This event-driven Monte Carlo algorithm is nonlocal and rejection free and allows for the breaking of detailed balance. The algorithm uses a discretized potential, but its running speed is asymptotically independent of the discretization. We apply the algorithm to two-dimensional soft spheres and discuss its possible implementation directly in the continuum limit.

Magnetism, entropy, and the first nano-machines

Abstract The efficiency of bio-molecular motors stems from reversible interactions ∼ k B T; weak bonds stabilizing intermediate states (enabling direct conversion of chemical into mechanical energy). For their (unknown) origins, we suggest that a magnetically structured phase (MSP) formed via accretion of superparamagnetic particles (S-PPs) during serpentinization (including magnetite formation) of igneous rocks comprising the Hadean Ocean floor, had hosted motor-like diffusion of ligand-bound S-PPs through its ‘template’-layers. Ramifications range from optical activity to quantum coherence. A gentle flux gradient offers both detailed-balance breaking non-equilibrium and asymmetry to a magnetic dipole, undergoing infinitesimal spin-alignment changes. Periodic perturbation of this background by local H-fields of templatepartners can lead to periodic high and low-template affinity states, due to the dipole’s magnetic degree of freedom. An accompanying magnetocaloric effect allows interchange between system-entropy and bath temperature. We speculate on a magnetic reproducer in a setting close to the submarine hydrothermal mound-scenario of Russell and coworkers that could evolve bio-ratchets.

Three Detailed Fluctuation Theorems

The total entropy production of a trajectory can be split into an adiabatic and a nonadiabatic contribution, deriving, respectively, from the breaking of detailed balance via nonequilibrium boundary conditions or by external driving. We show that each of them, the total, the adiabatic, and the nonadiabatic trajectory entropy, separately satisfies a detailed fluctuation theorem.

Cosmological perturbations in Horava-Lifshitz theory without detailed balance

In the Horava-Lifshitz theory of quantum gravity, two conditions -- detailed balance and projectability -- are usually assumed. The breaking of projectability simplifies the theory, but it leads to serious problems with the theory. The breaking of detailed balance leads to a more complicated form of the theory, but it appears to resolve some of the problems. Sotiriou, Visser and Weinfurtner formulated the most general theory of Horava-Lifshitz type without detailed balance. We compute the linear scalar perturbations of the FRW model in this form of HL theory. We show that the higher-order curvature terms in the action lead to a gravitational effective anisotropic stress on small scales. Specializing to a Minkowski background, we study the spin-0 scalar mode of the graviton, using a gauge-invariant analysis, and find that it is stable in both the infrared and ultraviolet regimes for $0 \le \xi \le 2/3$. However, in this parameter range the scalar mode is a ghost.

Entropy fluctuation theorems in driven open systems: Application to electron counting statistics

The total entropy production generated by the dynamics of an externally driven systems exchanging energy and matter with multiple reservoirs and described by a master equation is expressed as the sum of three contributions, each corresponding to a distinct mechanism for bringing the system out of equilibrium: Nonequilibrium initial conditions, external driving, and breaking of detailed balance. We derive three integral fluctuation theorems (FTs) for these contributions and show that they lead to the following universal inequality: An arbitrary nonequilibrium transformation always produces a change in the total entropy production greater than or equal to the one produced if the transformation is done very slowly (adiabatically). Previously derived fluctuation theorems can be recovered as special cases. We show how these FTs can be experimentally tested by performing the counting statistics of the electrons crossing a single level quantum dot coupled to two reservoirs with externally varying chemical potentials. The entropy probability distributions are simulated for driving protocols ranging from the adiabatic to the sudden switching limit.

Riemann-Cartan-Weyl geometries, quantum diffusions and the equivalence of the free maxwell and Dirac-Hestenes equations

We introduce the Riemann-Cartan-Weyl (RCW) space-time geometries of quantum mechanics with the most general trace-torsion non-exact Weyl 1-form, and characterize it in the Clifford bundle. Two electromagnetic potentials appear in the Weyl form, one having a zero field and the other one being the codifferential of a 2-form. We give the derivation of the non-linear equation for the wave function producing the exact Weyl one-form, which also defines the amplitude of a Dirac-Hestenes spinor operator field (DHSOF). We prove an equivalence between the free Maxwell equation for an extremal electromagnetic field and the Dirac-Hestenes equation for a DHSOF on a Riemann-Cartan-Weyl manifold, associating the electromagnetic potentials of the Weyl one-form with the internal electromagnetic potentials derived from the rotational dependance of a DHSOF. We show that this association produces a breaking of detailed balance in the spin plane. We discuss the relations with stochastic electrodynamics and the Navier-Stokes equation.

Formation of pyramids and mounds in molecular beam epitaxy.

One of the generic scenarios in molecular beam epitaxy is the growth of three-dimensional structures such as mounds and pyramids. These structures are a nonequilibrium effect thought to be due to a combination of the microscopic Ehrlich-Schwoebel barriers and the breaking of detailed balance by the deposition process. We propose and investigate by computer simulation a simple microscopic model that displays (i) slope selection, (ii) pyramid and moundlike structures, and (iii) coarsening. The characteristic length scale of our three-dimensional features grows as $R(t)\ensuremath{\sim}{t}^{n}$ with $n$ between 0.17 and 0.26. We discuss these results in the light of recent experiments and continuum models of molecular beam epitaxy.

Spatial correlations in reaction-diffusion systems in nonequilibrium conditions

The steady state spatial correlations in reaction-diffusion systems involving many reversible chemical reactions are examined. It has been already discussed that the spatial correlations are related to the breaking of detailed balance in chemical kinetics for both one species and for two species reversible reactions. Here, we focus our attention on how the spatial correlations of concentration fluctuations in a macroscopically homogeneous systems approach to the instability point. The spatial correlations depend strongly on the stability of systems for two species reactions compared to one species reactions.