Minimum entropy production, detailed balance and Wasserstein distance for continuous-time Markov processes

Journal of Physics A Mathematical and Theoretical ◽

10.1088/1751-8121/ac4ac0 ◽

2022 ◽

Author(s):

Andreas Dechant

Keyword(s):

Entropy Production ◽

Continuous Time ◽

Optimal Transport ◽

Detailed Balance ◽

Wasserstein Distance ◽

Minimum Entropy ◽

Markov Jump ◽

Continuous Space ◽

Minimum Entropy Production ◽

Spin Flips

Abstract We investigate the problem of minimizing the entropy production for a physical process that can be described in terms of a Markov jump dynamics. We show that, without any further constraints, a given time-evolution may be realized at arbitrarily small entropy production, yet at the expense of diverging activity. For a fixed activity, we find that the dynamics that minimizes the entropy production is given in terms of conservative forces. The value of the minimum entropy production is expressed in terms of the graph-distance based Wasserstein distance between the initial and final configuration. This yields a new kind of speed limit relating dissipation, the average number of transitions and the Wasserstein distance. It also allows us to formulate the optimal transport problem on a graph in term of a continuous-time interpolating dynamics, in complete analogy to the continuous space setting. We demonstrate our findings for simple state networks, a time-dependent pump and for spin flips in the Ising model.

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The principle of minimum entropy production and snow structure

Journal of Glaciology ◽

10.3189/172756504781829945 ◽

2004 ◽

Vol 50 (170) ◽

pp. 342-352 ◽

Cited By ~ 3

Author(s):

Perry Bartelt ◽

Othmar Buser

Keyword(s):

Mass Transfer ◽

Temperature Gradient ◽

Entropy Production ◽

Surface Curvature ◽

The State ◽

Temperature Gradients ◽

Curvature Radius ◽

Minimum Entropy ◽

Snow Layer ◽

Minimum Entropy Production

AbstractAn essential problem in snow science is to predict the changing form of ice grains within a snow layer. Present theories are based on the idea that form changes are driven by mass diffusion induced by temperature gradients within the snow cover. This leads to the well-established theory of isothermal- and temperature-gradient metamorphism. Although diffusion theory treats mass transfer, it does not treat the influence of this mass transfer on the form — the curvature radius of the grains and bonds — directly. Empirical relations, based on observations, are additionally required to predict flat or rounded surfaces. In the following, we postulate that metamorphism, the change of ice surface curvature and size, is a process of thermodynamic optimization in which entropy production is minimized. That is, there exists an optimal surface curvature of the ice grains for a given thermodynamic state at which entropy production is stationary. This state is defined by differences in ice and air temperature and vapor pressure across the interfacial boundary layer. The optimal form corresponds to the state of least wasted work, the state of minimum entropy production. We show that temperature gradients produce a thermal non-equilibrium between the ice and air such that, depending on the temperature, flat surfaces are required to mimimize entropy production. When the temperatures of the ice and air are equal, larger curvature radii are found at low temperatures than at high temperatures. Thus, what is known as isothermal metamorphism corresponds to minimum entropy production at equilibrium temperatures, and so-called temperature-gradient metamorphism corresponds to minimum entropy production at none-quilibrium temperatures. The theory is in good agreement with general observations of crystal form development in dry seasonal alpine snow.

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