We revisit the Ornstein-Uhlenbeck (OU) process as the fundamental mathematical description of lin... more We revisit the Ornstein-Uhlenbeck (OU) process as the fundamental mathematical description of linear irreversible phenomena, with fluctuations, near an equilibrium. By identifying the underlying circulating dynamics in a stationary process as the natural generalization of classical conservative mechanics, a bridge between a family of OU processes with equilibrium fluctuations and thermodynamics is established through the celebrated Helmholtz theorem. The Helmholtz theorem provides an emergent macroscopic “equation of state” of the entire system, which exhibits a universal ideal thermodynamic behavior. Fluctuating macroscopic quantities are studied from the stochastic thermodynamic point of view and a non-equilibrium work relation is obtained in the macroscopic picture, which may facilitate experimental study and application of the equalities due to Jarzynski, Crooks, and Hatano and Sasa.
We provide a constructive proof on the equivalence of two fundamental concepts: the global Lyapun... more We provide a constructive proof on the equivalence of two fundamental concepts: the global Lyapunov function in engineering and the potential function in physics, establishing a bridge between these distinct fields. This result suggests new approaches on the significant unsolved problem namely to construct Lyapunov functions for general nonlinear systems through the analogy with existing methods on potential functions. In addition, we show another connection that the Lyapunov equation is a reduced form of the generalized Einstein relation for linear systems. Comment: 4 pages
We find exact mappings for a class of limit cycle systems with noise onto quasi-symplectic dynami... more We find exact mappings for a class of limit cycle systems with noise onto quasi-symplectic dynamics, including a van der Pol type oscillator. A dual role potential function is obtained as a component of the quasi-symplectic dynamics. Based on a stochastic interpretation different from the traditional Ito's and Stratonovich's, we show the corresponding steady state distribution is the familiar Boltzmann-Gibbs type for arbitrary noise strength. The result provides a new angle for understanding processes without detailed balance and can be verified by experiments.
ABSTRACT One of the present authors has proposed a novel treatment of stochastic processes leadin... more ABSTRACT One of the present authors has proposed a novel treatment of stochastic processes leading to the construction of potential functions for dynamics described by stochastic differential equations (SDEs). The approach transforms the deterministic part of the original system into three components: a potential function, a frictional force and a Lorentz force. The potential function drives the dynamics and determines the final steady state distribution that has both local and global meaning. We note that such a potential is closely related to the classical Lyapunov function. In this paper, we first provide a brief review on the decomposition framework and then give a constructive proof on the equivalence of two fundamental concepts: the global Lyapunov function in engineering and the potential function in physics, establishing a bridge between the two distinct fields. This result reveals the physical meaning of Lyapunov functions, thus suggests new approaches on the largely unsolved problem: constructing Lyapunov functions for general nonlinear systems, through the analogy with existing methods on potential functions. In addition, we show another connection that the Lyapunov equation is a reduced form of the generalized Einstein relation for linear systems. By inheriting from a physical treatment of stochastic processes, this work demonstrates a stochastic view of deterministic systems together with the deterministic rules that govern stochastic behaviors.
2013 22nd International Conference on Noise and Fluctuations (ICNF), 2013
ABSTRACT An energy potential function is constructed for a Lorenz-like piecewise continuous dissi... more ABSTRACT An energy potential function is constructed for a Lorenz-like piecewise continuous dissipative system. Using this nonincreasing potential function, the system's dynamics is decomposed into a gradient component, accounting for energy dissipation; and a rotation component, facilitating energy conserved motion. Each part has an individual role of determining the strangeness or chaoticness of the attractor.
Based on conventional Ito or Stratonovich interpretation, zero-mean multiplicative noise can indu... more Based on conventional Ito or Stratonovich interpretation, zero-mean multiplicative noise can induce shifts of attractors or even changes of topology to a deterministic dynamics. Such phenomena usually introduce additional complications in analysis of these systems. We employ in this paper a new stochastic interpretation leading to a straightforward consequence: The steady state distribution is Boltzmann-Gibbs type with a potential function severing as a Lyapunov function for the deterministic dynamics. It implies that an attractor corresponds to the local extremum of the distribution function and the probability is equally distributed right on an attractor. We consider a prototype of nonequilibrium processes, noisy limit cycle dynamics. Exact results are obtained for a class of limit cycles, including a van der Pol type oscillator. These results provide a new angle for understanding processes without detailed balance and can be verified by experiments.
We revisit the Ornstein-Uhlenbeck (OU) process as the fundamental mathematical description of lin... more We revisit the Ornstein-Uhlenbeck (OU) process as the fundamental mathematical description of linear irreversible phenomena, with fluctuations, near an equilibrium. By identifying the underlying circulating dynamics in a stationary process as the natural generalization of classical conservative mechanics, a bridge between a family of OU processes with equilibrium fluctuations and thermodynamics is established through the celebrated Helmholtz theorem. The Helmholtz theorem provides an emergent macroscopic “equation of state” of the entire system, which exhibits a universal ideal thermodynamic behavior. Fluctuating macroscopic quantities are studied from the stochastic thermodynamic point of view and a non-equilibrium work relation is obtained in the macroscopic picture, which may facilitate experimental study and application of the equalities due to Jarzynski, Crooks, and Hatano and Sasa.
We provide a constructive proof on the equivalence of two fundamental concepts: the global Lyapun... more We provide a constructive proof on the equivalence of two fundamental concepts: the global Lyapunov function in engineering and the potential function in physics, establishing a bridge between these distinct fields. This result suggests new approaches on the significant unsolved problem namely to construct Lyapunov functions for general nonlinear systems through the analogy with existing methods on potential functions. In addition, we show another connection that the Lyapunov equation is a reduced form of the generalized Einstein relation for linear systems. Comment: 4 pages
We find exact mappings for a class of limit cycle systems with noise onto quasi-symplectic dynami... more We find exact mappings for a class of limit cycle systems with noise onto quasi-symplectic dynamics, including a van der Pol type oscillator. A dual role potential function is obtained as a component of the quasi-symplectic dynamics. Based on a stochastic interpretation different from the traditional Ito's and Stratonovich's, we show the corresponding steady state distribution is the familiar Boltzmann-Gibbs type for arbitrary noise strength. The result provides a new angle for understanding processes without detailed balance and can be verified by experiments.
ABSTRACT One of the present authors has proposed a novel treatment of stochastic processes leadin... more ABSTRACT One of the present authors has proposed a novel treatment of stochastic processes leading to the construction of potential functions for dynamics described by stochastic differential equations (SDEs). The approach transforms the deterministic part of the original system into three components: a potential function, a frictional force and a Lorentz force. The potential function drives the dynamics and determines the final steady state distribution that has both local and global meaning. We note that such a potential is closely related to the classical Lyapunov function. In this paper, we first provide a brief review on the decomposition framework and then give a constructive proof on the equivalence of two fundamental concepts: the global Lyapunov function in engineering and the potential function in physics, establishing a bridge between the two distinct fields. This result reveals the physical meaning of Lyapunov functions, thus suggests new approaches on the largely unsolved problem: constructing Lyapunov functions for general nonlinear systems, through the analogy with existing methods on potential functions. In addition, we show another connection that the Lyapunov equation is a reduced form of the generalized Einstein relation for linear systems. By inheriting from a physical treatment of stochastic processes, this work demonstrates a stochastic view of deterministic systems together with the deterministic rules that govern stochastic behaviors.
2013 22nd International Conference on Noise and Fluctuations (ICNF), 2013
ABSTRACT An energy potential function is constructed for a Lorenz-like piecewise continuous dissi... more ABSTRACT An energy potential function is constructed for a Lorenz-like piecewise continuous dissipative system. Using this nonincreasing potential function, the system's dynamics is decomposed into a gradient component, accounting for energy dissipation; and a rotation component, facilitating energy conserved motion. Each part has an individual role of determining the strangeness or chaoticness of the attractor.
Based on conventional Ito or Stratonovich interpretation, zero-mean multiplicative noise can indu... more Based on conventional Ito or Stratonovich interpretation, zero-mean multiplicative noise can induce shifts of attractors or even changes of topology to a deterministic dynamics. Such phenomena usually introduce additional complications in analysis of these systems. We employ in this paper a new stochastic interpretation leading to a straightforward consequence: The steady state distribution is Boltzmann-Gibbs type with a potential function severing as a Lyapunov function for the deterministic dynamics. It implies that an attractor corresponds to the local extremum of the distribution function and the probability is equally distributed right on an attractor. We consider a prototype of nonequilibrium processes, noisy limit cycle dynamics. Exact results are obtained for a class of limit cycles, including a van der Pol type oscillator. These results provide a new angle for understanding processes without detailed balance and can be verified by experiments.
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