Fokker-Planck Equation and Thermodynamic System Analysis

Lucia, Umberto; Gervino, G.

doi:10.3390/e17020763

Cited by 6 publications

(6 citation statements)

References 25 publications

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“…The boundary condition that we impose will be introduced in section 2. FPE is a fundamental problem in the literature of statistical physics due to its wide applications in thermodynamic system analysis (Markowich and Villani, 2000;Lucia and Gervino, 2015;Qi and Majda, 2016) and is one of the key equations in the research of the mean field game (Cardaliaguet and Porretta, 2020;Gomes et al, 2014). Recently, it has also been used to model the dynamics of the stochastic gradient descent method on neural networks (Chizat and Bach, 2018;Sonoda and Murata, 2019;Sirignano and Spiliopoulos, 2020;Fang et al, 2021) and the dynamics of the Rényi differential privacy (Chourasia et al, 2021), and has become a fundamental tool for learning complex distributions and deep generative models due to its deep connection to the Wasserstein gradient flow (Sohl-Dickstein et al, 2015;Hashimoto et al, 2016;Liu et al, 2019;Solin et al, 2021;Mokrov et al, 2021).…”

Section: Introductionmentioning

confidence: 99%

Self-Consistency of the Fokker-Planck Equation

Shen¹,

Wang²,

Kale³

et al. 2022

Preprint

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The Fokker-Planck equation (FPE) is the partial differential equation that governs the density evolution of the Itô process and is of great importance to the literature of statistical physics and machine learning. The FPE can be regarded as a continuity equation where the change of the density is completely determined by a time varying velocity field. Importantly, this velocity field also depends on the current density function. As a result, the ground-truth velocity field can be shown to be the solution of a fixed-point equation, a property that we call self-consistency. In this paper, we exploit this concept to design a potential function of the hypothesis velocity fields, and prove that, if such a function diminishes to zero during the training procedure, the trajectory of the densities generated by the hypothesis velocity fields converges to the solution of the FPE in the Wasserstein-2 sense. The proposed potential function is amenable to neural-network based parameterization as the stochastic gradient with respect to the parameter can be efficiently computed. Once a parameterized model, such as Neural Ordinary Differential Equation is trained, we can generate the entire trajectory to the FPE.

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Section: Introductionmentioning

confidence: 99%

Self-Consistency of the Fokker-Planck Equation

Shen¹,

Wang²,

Kale³

et al. 2022

Preprint

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“…Here a solution to this paradox is introduced to extend the use of the entropy generation to small systems thus obtaining a link between local and global thermodynamic quantities. Moreover, the use of this new approach avoids the difficulties highlighted in the usual analysis of the small systems, such as the definition of a temperature for nanosystems, even if some thermodynamicists have began the analysis of this topic [93,94,105,108]. Moreover, in this paper we have only introduced a first application of the GSGL approach to nanosystems.…”

Section: Discussionmentioning

confidence: 99%

A Link between Nano- and Classical Thermodynamics: Dissipation Analysis (The Entropy Generation Approach in Nano-Thermodynamics)

Lucia

2015

Entropy

Self Cite

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Abstract:The interest in designing nanosystems is continuously growing. Engineers apply a great number of optimization methods to design macroscopic systems. If these methods could be introduced into the design of small systems, a great improvement in nanotechnologies could be achieved. To do so, however, it is necessary to extend classical thermodynamic analysis to small systems, but irreversibility is also present in small systems, as the Loschmidt paradox highlighted. Here, the use of the recent improvement of the Gouy-Stodola theorem to complex systems (GSGL approach), based on the use of entropy generation, is suggested to obtain the extension of classical thermodynamics to nanothermodynamics. The result is a new approach to nanosystems which avoids the difficulties highlighted in the usual analysis of the small systems, such as the definition of temperature for nanosystems.

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“…Also, the well-known quantum uncertainty relation can be proven to hold for non-quantum but stochastic trajectories of a Brownian particle [25]. Furthermore, the entropy generation during the stochastic evolution of a system has been studied by means of the Gouy-Stodola theorem [26] and applied to the biological context to model molecular machines in the orginal way [27] and to study the control and regulation of temperature in cells [28]. Other approaches for investigating the behavior of molecular motors are based on the over-damped Langevin equation and have been successfully compared to the experimental data of the F 1 -ATPase motor [29,30].…”

Section: Introductionmentioning

confidence: 99%

Full Statistics of Conjugated Thermodynamic Ensembles in Chains of Bistable Units

Benedito

Manca²,

Giordano

2019

Inventions

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The statistical mechanics and the thermodynamics of small systems are characterized by the non-equivalence of the statistical ensembles. When concerning a polymer chain or an arbitrary chain of independent units, this concept leads to different force-extension responses for the isotensional (Gibbs) and the isometric (Helmholtz) thermodynamic ensembles for a limited number of units (far from the thermodynamic limit). While the average force-extension response has been largely investigated in both Gibbs and Helmholtz ensembles, the full statistical characterization of this thermo-mechanical behavior has not been approached by evaluating the corresponding probability densities. Therefore, we elaborate in this paper a technique for obtaining the probability density of the extension when force is applied (Gibbs ensemble) and the probability density of the force when the extension is prescribed (Helmholtz ensemble). This methodology, here developed at thermodynamic equilibrium, is applied to a specific chain composed of units characterized by a bistable potential energy, which is able to mimic the folding and unfolding of several macromolecules of biological origin.

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Fokker-Planck Equation and Thermodynamic System Analysis

Cited by 6 publications

References 25 publications

Self-Consistency of the Fokker-Planck Equation

Self-Consistency of the Fokker-Planck Equation

A Link between Nano- and Classical Thermodynamics: Dissipation Analysis (The Entropy Generation Approach in Nano-Thermodynamics)

Full Statistics of Conjugated Thermodynamic Ensembles in Chains of Bistable Units

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