Potentials of continuous Markov processes and random perturbations

Yang, Ying-Jen; Cheng, Yu-Chen

doi:10.1088/1751-8121/abef80

Cited by 11 publications

(9 citation statements)

References 35 publications

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“…where b ρ,J := −D∇ (− ln ρ) + J/ρ is vector field decomposition from stationary distribution and flux [75], note that J is divergence free so b ρ,J can be further rewritten in terms of its bivector potential [62,76]. The alternative form of Eq.…”

Section: Discussionmentioning

confidence: 99%

Statistical Thermodynamics and Data Infinitum: Conjugate Variables as Forces, and their Statistical Variations

Yang¹,

Qian²

2022

Preprint

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Following ideas of Szilard, Mandelbrot and Hill, we show that a statistical thermodynamic structure can emerge purely from the infinitely large data limit under a probabilistic framework independent from their underlying details. Systems with distinct values of a set of observables are identified as different thermodynamic states, which are parameterized by the entropic forces conjugated to the observables. The ground state with zero entropic forces usually has a probabilistic model equipped with a symmetry of interest. The entropic forces lead to symmetry breaking for each particular system that produces the data, cf. emerging of time correlation and breakdown of detailed balance. Probabilistic models for the excited states are predicted by the Maximum Entropy Principle for sequences of i.i.d. and correlated Markov samples. Asymptotically-equivalent models are also found by the Maximum Caliber Principle. With a novel derivation of Maximum Caliber, conceptual differences between the two principles are clarified. The emergent thermodynamics in the data infinitus limit has a mesoscopic origin from the Maximum Caliber. In the canonical probabilistic models of Maximum Caliber, the variances of the observables and their conjugated forces satisfy the asymptotic thermodynamic uncertainty principle, which stems from the reciprocal-curvature relation between "entropy" and "free energy" functions in the theory of large deviations. The mesoscopic origin of the reciprocality is identified. As a consequence of limit theorems in probability theory, the phenomenological statistical thermodynamics is universal without the need of mechanics.

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Section: Discussionmentioning

confidence: 99%

Statistical Thermodynamics and Data Infinitum: Conjugate Variables as Forces, and their Statistical Variations

Yang¹,

Qian²

2022

Preprint

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“…My Ph.D. group has a tradition of working on thermodynamics of stochastic processes [13,38,17,18,37,4,5]. I worked on a problem of comparing the thermodynamic properties of a Markov chain and its covering space.…”

Section: Thermodynamics In Stochastic Processesmentioning

confidence: 99%

Finding meaningful and workable applied mathematics problems in science

Wang¹

2023

Preprint

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In this short review, I will summarize my research experience in three fields in applied mathematics: mathematical biology, applied probability, and applied discrete mathematics. Specifically, I will show how each project was initiated, and what wrong approaches were applied. Such details are important in learning how to do research, but they cannot be read out from research papers. I wish that students and junior researchers in applied mathematics could learn a lesson from this summary.

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“…For most of the applications, one solves (i) based on (ii) and (iii). Alternatively, given (i) and (iii), (ii) admits a general decomposition in terms of the other two [16][17][18]. All these previous results are now encompassed in the GER (Eq.…”

mentioning

confidence: 93%

“…The vector field decomposition b = −D∇Φ − Q∇Φ + ∇ × Q discussed in Ref. [18] provides an alternative proof for the GER in continuous processes. By using integration by part,…”

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confidence: 99%

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Nonequilibrium Stationary Process and Fluctuation-Dissipation Relations

Yang,

Qian

2021

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A stochastic dynamics has a natural decomposition into a drift capturing mean rate of change and a martingale increment capturing randomness. They are two statistically uncorrelated, but not necessarily independent mechanisms contributing to the overall fluctuations of the dynamics, representing the uncertainties in the past and in the future. A generalized Einstein relation is a consequence solely because the dynamics being stationary; and the Green-Kubo formula reflects a balance between the two mechanisms. Equilibrium with reversibility is characterized by a novel covariance symmetry.

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Potentials of continuous Markov processes and random perturbations

Cited by 11 publications

References 35 publications

Statistical Thermodynamics and Data Infinitum: Conjugate Variables as Forces, and their Statistical Variations

Statistical Thermodynamics and Data Infinitum: Conjugate Variables as Forces, and their Statistical Variations

Finding meaningful and workable applied mathematics problems in science

Nonequilibrium Stationary Process and Fluctuation-Dissipation Relations

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