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Hong Qian

Publications and source records attributed to Hong Qian.

At least 73 records · Page 4Linked to original sources

Derivative-Free Reinforcement Learning: A Review

Reinforcement learning is about learning agent models that make the best sequential decisions in unknown environments. In an unknown environment, the agent needs to explore the environment while exploiting the collected information, which usually forms a sophisticated problem to solve. Derivative-free optimization, meanwhile, is capable of solving sophisticated problems. It commonly uses a sampling-and-updating framework to iteratively improve the solution, where exploration and exploitation are also needed to be well balanced. Therefore, derivative-free optimization deals with a similar core issue as reinforcement learning, and has been introduced in reinforcement learning approaches, under the names of learning classifier systems and neuroevolution/evolutionary reinforcement learning. Although such methods have been developed for decades, recently, derivative-free reinforcement learning exhibits attracting increasing attention. However, recent survey on this topic is still lacking. In this article, we summarize methods of derivative-free reinforcement learning to date, and organize the methods in aspects including parameter updating, model selection, exploration, and parallel/distributed methods. Moreover, we discuss some current limitations and possible future directions, hoping that this article could bring more attentions to this topic and serve as a catalyst for developing novel and efficient approaches.

cs.LG↗

Asymptotic Behavior of a Sequence of Conditional Probability Distributions and the Canonical Ensemble

The probability distribution of a function of a subsystem conditioned on the value of the function of the whole, in the limit when the ratio of their values goes to zero, has a limit law: It equals the unconditioned marginal probability distribution weighted by an exponential factor whose exponent is uniquely determined by the condition. We apply this theorem to explain the canonical equilibrium ensemble of a system in contact with a heat reservoir. Since the theorem only requires analysis at the level of the function of the subsystem and reservoir, it is applicable even without the knowledge of the composition of the reservoir itself, which extends the applicability of the canonical ensemble. Furthermore, we generalize our theorem to a model with strong interaction that contributes an additional term to the exponent, which is beyond the typical case of approximately additive functions. This result is new in both physics and mathematics, as a theory for the Gibbs conditioning principle for strongly correlated systems. A corollary provides a precise formulation of what a temperature bath is in probabilistic term

math-ph↗

Bivectorial Nonequilibrium Thermodynamics: Cycle Affinity, Vorticity Potential, and Onsager's Principle

We generalize an idea in the works of Landauer and Bennett on computations, and Hill's in chemical kinetics, to emphasize the importance of kinetic cycles in mesoscopic nonequilibrium thermodynamics (NET). For continuous stochastic systems, a NET in phase space is formulated in terms of cycle affinity $\nabla\wedge\big(\mathbf{D}^{-1}\mathbf{b}\big)$ and vorticity potential $\mathbf{A}(\mathbf{x})$ of the stationary flux $\mathbf{J}^{*}=\nabla\times\mathbf{A}$. Each bivectorial cycle couples two transport processes represented by vectors and gives rise to Onsager's reciprocality; the scalar product of the two bivectors $\mathbf{A}\cdot\nabla\wedge\big(\mathbf{D}^{-1}\mathbf{b}\big)$ is the rate of local entropy production in the nonequilibrium steady state. An Onsager operator that maps vorticity to cycle affinity is introduced.

cond-mat.stat-mech↗

Stochastic limit-cycle oscillations of a nonlinear system under random perturbations

Dynamical systems with $ε$ small random perturbations appear in both continuous mechanical motions and discrete stochastic chemical kinetics. The present work provides a detailed analysis of the central limit theorem (CLT), with a time-inhomogeneous Gaussian process, near a deterministic limit cycle in $\mathbb{R}^n$. Based on the theory of random perturbations of dynamical systems and the WKB approximation respectively, results are developed in parallel from both standpoints of stochastic trajectories and transition probability density and their relations are elucidated. We show rigorously the correspondence between the local Gaussian fluctuations and the curvature of the large deviation rate function near its infimum, connecting the CLT and the large deviation principle of diffusion processes. We study uniform asymptotic behavior of stochastic limit cycles through the interchange of limits of time $t\to\infty$ and $ε\to 0$. Three further characterizations of stochastic limit cycle oscillators are obtained: (i) An approximation of the probability flux near the cycle; (ii) Two special features of the vector field for the cyclic motion; (iii) A local entropy balance equation along the cycle with clear physical meanings. Lastly and different from the standard treatment, the origin of the $ε$ in the theory is justified by a novel scaling hypothesis via constructing a sequence of stochastic differential equations.

math-ph↗

Representations and Divergences in the Space of Probability Measures and Stochastic Thermodynamics

Radon-Nikodym (RN) derivative between two measures arises naturally in the affine structure of the space of probability measures with densities. Entropy, free energy, relative entropy, and entropy production as mathematical concepts associated with RN derivatives are introduced. We identify a simple equation that connects two measures with densities as a possible mathematical basis of the entropy balance equation that is central in nonequilibrium thermodynamics. Application of this formalism to Gibbsian canonical distribution yields many results in classical thermomechanics. An affine structure based on the canonical represenation and two divergences are introduced in the space of probability measures. It is shown that thermodynamic work, as a conditional expectation, is indictive of the RN derivative between two energy represenations being singular. The entropy divergence and the heat divergence yield respectively a Massieu-Planck potential based and a generalized Carnot inequalities.

cond-mat.stat-mech↗

Thermodynamics of Markov Processes with Non-extensive Entropy and Free Energy

Statistical thermodynamics of small systems shows dramatic differences from normal systems. Parallel to the recently presented steady-state thermodynamic formalism for master equation and Fokker-Planck equation, we show that a ``thermodynamic'' theory can also be developed based on Tsallis' generalized entropy $S^{(q)}=\sum_{i=1}^N(p_i-p_i^q)/[q(q-1)]$ and Shiino's generalized free energy $F^{(q)}=[\sum_{i=1}^Np_i(p_i/π_i)^{q-1}-1]/[q(q-1)]$, where $π_i$ is the stationary distribution. $dF^{(q)}/dt=-f_d^{(q)}\le 0$ and it is zero iff the system is in its stationary state. $dS^{(q)}/dt-Q_{ex}^{(q)} = f_d^{(q)}$ where $Q_{ex}^{(q)}$ characterizes the heat exchange. For systems approaching equilibrium with detailed balance, $f_d^{(q)}$ is the product of Onsager's thermodynamic flux and force. However, it is discovered that the Onsager's force is non-local. This is a consequence of the particular transformation invariance for zero energy of Tsallis' statistics.

math-ph↗

Kinematic Basis of Emergent Energetics of Complex Dynamics

Stochastic kinematic description of a complex dynamics is shown to dictate an energetic and thermodynamic structure. An energy function $φ(x)$ emerges as the limit of the generalized, nonequilibrium free energy of a Markovian dynamics with vanishing fluctuations. In terms of the $\nablaφ$ and its orthogonal field $γ(x)\perp\nablaφ$, a general vector field $b(x)$ can be decomposed into $-D(x)\nablaφ+γ$, where $\nabla\cdot\big(ω(x)γ(x)\big)=$ $-\nablaωD(x)\nablaφ$. The matrix $D(x)$ and scalar $ω(x)$, two additional characteristics to the $b(x)$ alone, represent the local geometry and density of states intrinsic to the statistical motion in the state space at $x$. $φ(x)$ and $ω(x)$ are interpreted as the emergent energy and degeneracy of the motion, with an energy balance equation $dφ(x(t))/dt=γD^{-1}γ-bD^{-1}b$, reflecting the geometrical $\|D\nablaφ\|^2+\|γ\|^2=\|b\|^2$. The partition function employed in statistical mechanics and J. W. Gibbs' method of ensemble change naturally arise; a fluctuation-dissipation theorem is established via the two leading-order asymptotics of entropy production as $ε\to 0$. The present theory provides a mathematical basis for P. W. Anderson's emergent behavior in the hierarchical structure of complexity science.

physics.gen-ph↗

The Statistical Foundation of Entropy in Extended Irreversible Thermodynamics

In the theory of extended irreversible thermodynamics (EIT), the flux-dependent entropy function plays a key role and has a fundamental distinction from the usual flux-independent entropy function adopted by classical irreversible thermodynamics (CIT). However, its existence, as a prerequisite for EIT, and its statistical origin have never been justified. In this work, by studying the macroscopic limit of an ε-dependent Langevin dynamics, which admits a large deviations (LD) principle, we show that the stationary LD rate functions of probability density p_ε(x; t) and joint probability density p_ε(x; \dot{x}; t) actually turn out to be the desired fluxindependent entropy function in CIT and flux-dependent entropy function in EIT respectively. The difference of the two entropy functions is determined by the time resolution for Brownian motions times a Lagrangian, the latter arises from the LD Hamilton-Jacobi equation and can be used for constructing conserved Lagrangian/Hamiltonian dynamics.

math.PR↗

Unified formalism for entropy productions and fluctuation relations

Stochastic entropy production, which quantifies the difference between the probabilities of trajectories of a stochastic dynamics and its time reversals, has a central role in nonequilibrium thermodynamics. In the theory of probability, the change in the statistical properties of observables can be represented by a change in the probability measure. We consider operators on the space of probability measure that induce changes in the statistical properties of a process, and formulate entropy productions in terms of these change-of-probability-measure (CPM) operators. This mathematical underpinning of the origin of entropy productions allows us to achieve an organization of various forms of fluctuation relations: All entropy productions have a non-negative mean value, admit the integral fluctuation theorem, and satisfy a rather general fluctuation relation. Other results such as the transient fluctuation theorem and detailed fluctuation theorems then are derived from the general fluctuation relation with more constraints on the operator. We use a discrete-time, discrete-state-space Markov process to draw the contradistinction among three reversals of a process: time reversal, protocol reversal and the dual process. The properties of their corresponding CPM operators are examined, and the domains of validity of various fluctuation relations for entropy productions in physics and chemistry are revealed. We also show that our CPM operator formalism can help us rather easily extend other fluctuations relations for excess work and heat, discuss the martingale properties of entropy productions, and derive the stochastic integral formulas for entropy productions in constant-noise diffusion process with Girsanov theorem. Our formalism provides a general and concise way to study the properties of entropy-related quantities in stochastic thermodynamics and information theory.

cond-mat.stat-mech↗

Universal Relation Between Thermodynamic Driving Force and One-Way Fluxes in a Nonequilibrium Chemical Reaction with Complex Mechanism

In nonequilibrium chemical reaction systems, a fundamental relationship between unbalanced kinetic one-way fluxes and thermodynamic chemical driving forces is believed to exists. However this relation has been rigorously demonstrated only in a few cases in which one-way fluxes are well defined. In terms of its stochastic kinetic representation, we formulate the one-way fluxes for a general chemical reaction far from equilibrium, with arbitrary complex mechanisms, multiple intermediates, and internal kinetic cycles. For each kinetic cycle, the logarithm of the ratio of the steady-state forward and backward one-way fluxes is equal to the free energy difference between the reactants and products along the cycle. This fundamental relation is further established for general chemical reaction networks with multiple input and output complexes. Our result not only provides an equivalent definition of free energy difference in nonequilibrium chemical reaction networks, it also unifies the stochastic and macroscopic nonequilibrium chemical thermodynamics in a very broad sense.

physics.chem-ph↗

Synchronization in discrete-time, discrete-state Random Dynamical Systems

We characterize synchronization phenomenon in discrete-time, discrete-state random dynamical systems, with random and probabilistic Boolean networks as particular examples. In terms of multiplicative ergodic properties of the induced linear cocycle, we show such a random dynamical system with finite state synchronizes if and only if the Lyapunov exponent $0$ has simple multiplicity. For the case of countable state space, characterization of synchronization is provided in terms of the spectral subspace corresponding to the Lyapunov exponent $-\infty$. In addition, for both cases of finite and countable state spaces, the mechanism of partial synchronization is described by partitioning the state set into synchronized subsets. Applications to biological networks are also discussed.

math.DS↗

Ternary Representation of Stochastic Change and the Origin of Entropy and Its Fluctuations

A change in a stochastic system has three representations: Probabilistic, statistical, and informational: (i) is based on random variable $u(ω)\to\tilde{u}(ω)$; this induces (ii) the probability distributions $F_u(x)\to F_{\tilde{u}}(x)$, $x\in\mathbb{R}^n$; and (iii) a change in the probability measure $\mathbb{P}\to\tilde{\mathbb{P}}$ under the same observable $u(ω)$. In the informational representation a change is quantified by the Radon-Nikodym derivative $\ln\left( \frac{ d \tilde{\mathbb{P}}}{ d\mathbb{P}}(ω)\right)=-\ln\left(\frac{ d F_u}{ d F_{\tilde{u}}}(x)\right)$ when $x=u(ω)$. Substituting a random variable into its own density function creates a fluctuating entropy whose expectation has been given by Shannon. Informational representation of a deterministic transformation on $\mathbb{R}^n$ reveals entropic and energetic terms, and the notions of configurational entropy of Boltzmann and Gibbs, and potential of mean force of Kirkwood. Mutual information arises for correlated $u(ω)$ and $\tilde{u}(ω)$; and a nonequilibrium thermodynamic entropy balance equation is identified.

cond-mat.stat-mech↗

Grand canonical diffusion-influenced reactions: a stochastic theory with applications to multiscale reaction-diffusion simulations

Smoluchowski-type models for diffusion-influenced reactions (A+B -> C) can be formulated within two frameworks: the probabilistic-based approach for a pair A, B of reacting particles and the concentration-based approach for systems in contact with a bath that generates a concentration gradient of B particles that interact with A. Although these two approaches are mathematically similar, it is not straightforward to establish a precise mathematical relationship between them. Determining this relationship is essential to derive particle-based numerical methods that are quantitatively consistent with bulk concentration dynamics. In this work, we determine the relationship between the two approaches by introducing the grand canonical Smoluchowski master equation (GC-SME), which consists of a continuous-time Markov chain that models an arbitrary number of B particles, each one of them following Smoluchowski's probabilistic dynamics. We show that the GC-SME recovers the concentration-based approach by taking either the hydrodynamic or the large copy number limit. In addition, we show that the GC-SME provides a clear statistical mechanical interpretation of the concentration-based approach and yields an emergent chemical potential for nonequilibrium spatially inhomogeneous reaction processes. We further exploit the GC-SME robust framework to accurately derive multiscale/hybrid numerical methods that couple particle-based reaction-diffusion simulations with bulk concentration descriptions, as described in detail through two computational implementations.

physics.chem-ph↗

Time Dependent Saddle Node Bifurcation: Breaking Time and the Point of No Return in a Non-Autonomous Model of Critical Transitions

There is a growing awareness that catastrophic phenomena in biology and medicine can be mathematically represented in terms of saddle-node bifurcations. In particular, the term `tipping', or critical transition has in recent years entered the discourse of the general public in relation to ecology, medicine, and public health. The saddle-node bifurcation and its associated theory of catastrophe as put forth by Thom and Zeeman has seen applications in a wide range of fields including molecular biophysics, mesoscopic physics, and climate science. In this paper, we investigate a simple model of a non-autonomous system with a time-dependent parameter $p(τ)$ and its corresponding `dynamic' (time-dependent) saddle-node bifurcation by the modern theory of non-autonomous dynamical systems. We show that the actual point of no return for a system undergoing tipping can be significantly delayed in comparison to the {\em breaking time} $\hatτ$ at which the corresponding autonomous system with a time-independent parameter $p_{a}= p(\hatτ)$ undergoes a bifurcation. A dimensionless parameter $α=λp_0^3V^{-2}$ is introduced, in which $λ$ is the curvature of the autonomous saddle-node bifurcation according to parameter $p(τ)$, which has an initial value of $p_{0}$ and a constant rate of change $V$. We find that the breaking time $\hatτ$ is always less than the actual point of no return $τ^*$ after which the critical transition is irreversible; specifically, the relation $τ^*-\hatτ\simeq 2.338(λV)^{-\frac{1}{3}}$ is analytically obtained. For a system with a small $λV$, there exists a significant window of opportunity $(\hatτ,τ^*)$ during which rapid reversal of the environment can save the system from catastrophe.

q-bio.QM↗

Stochastic Dynamics II: Finite Random Dynamical Systems, Linear Representation, and Entropy Production

We study finite state random dynamical systems (RDS) and their induced Markov chains (MC) as stochastic models for complex dynamics. The linear representation of deterministic maps in RDS are matrix-valued random variables whose expectations correspond to the transition matrix of the MC. The instantaneous Gibbs entropy, Shannon-Khinchin entropy per step, and the entropy production rate of the MC are discussed. These three concepts as key anchor points in stochastic dynamics, characterize respectively the uncertainties of the system at instant time $t$, the randomness generated per step, and the dynamical asymmetry with respect to time reversal. The entropy production rate, expressed in terms of the cycle distributions, has found an expression in terms of the probability of the deterministic maps with the single attractor in the maximum entropy RDS. For finite RDS with invertible transformations, the non-negative entropy production rate of its MC is bounded above by the Kullback-Leibler divergence of the probability of the deterministic maps with respect to its time-reversal dual probability.

math.DS↗

Both cellular ATP level and ATP hydrolysis free energy determine energetically the calcium oscillation in pancreatic $β$-cell

In pancreatic $β$-cells, calcium oscillation signal is the core part of glucose-stimulated insulin secretion. Intracellular calcium concentration oscillates in response to the intake of glucose, which triggers the exocytosis of insulin secretory granules. ATP plays a crucial part in this process. ATP increases as the result of glucose intake, then ATP binds to ATP-sensitive $K^+$ channels ($K_{ATP}$), depolarizes the cell and triggers calcium oscillation, while the ion pumps on the cell membrane consumes the free energy form ATP hydrolysis. Based on Betram et. al. 2004 model, we construct a kinetic models to analyze the thermodynamic characteristics of this system, to reveal how the ATP hydrolysis free energy affects the calcium oscillation. Our results suggest that bifurcation point is sensitive to both the free energy level and cellular ATP level, and the insufficient ATP energy supply would cause dysfunction of calcium oscillation.

q-bio.CB↗

Relaxation rates of gene expression kinetics reveal the feedback signs of autoregulatory gene networks

The transient response to a stimulus and subsequent recovery to a steady state are the fundamental characteristics of a living organism. Here we study the relaxation kinetics of autoregulatory gene networks based on the chemical master equation model of single-cell stochastic gene expression with nonlinear feedback regulation. We report a novel relation between the rate of relaxation, characterized by the spectral gap of the Markov model, and the feedback sign of the underlying gene circuit. When a network has no feedback, the relaxation rate is exactly the decaying rate of the protein. We further show that positive feedback always slows down the relaxation kinetics while negative feedback always speeds it up. Numerical simulations demonstrate that this relation provides a possible method to infer the feedback topology of autoregulatory gene networks by using time-series data of gene expression.

q-bio.MN↗

Dynamic Looping of a Free-Draining Polymer

We revisit the celebrated Wilemski-Fixman (WF) treatment for the looping time of a free-draining polymer. The WF theory introduces a sink term into the Fokker-Planck equation for the $3(N+1)$-dimensional Ornstein-Uhlenbeck process of the polymer dynamics, which accounts for the appropriate boundary condition due to the formation of a loop. The assumption for WF theory is considerably relaxed. A perturbation method approach is developed that justifies and generalizes the previous results using either a Delta sink or a Heaviside sink. For both types of sinks, we show that under the condition of a small dimensionless $ε$, the ratio of capture radius to the Kuhn length, we are able to systematically produce all known analytical and asymptotic results obtained by other methods. This includes most notably the transition regime between the $N^2$ scaling of Doi, and $N\sqrt{N}/ε$ scaling of Szabo, Schulten, and Schulten. The mathematical issue at play is the non-uniform convergence of $ε\to 0$ and $N\to\infty$, the latter being an inherent part of the theory of a Gaussian polymer. Our analysis yields a novel term in the analytical expression for the looping time with small $ε$, which is previously unknown. Monte Carlo numerical simulations corroborate the analytical findings. The systematic method developed here can be applied to other systems modeled by multi-dimensional Smoluchowski equations.

cond-mat.soft↗