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The Principle of Minimal Resistance in Non-equilibrium Thermodynamics

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Abstract

Analytical models describing the motion of colloidal particles in given force fields are presented. In addition to local approaches, leading to well known master equations such as the Langevin and the Fokker–Planck equations, a global description based on path integration is reviewed. A new result is presented, showing that under very broad conditions, during its evolution a dissipative system tends to minimize its energy dissipation in such a way to keep constant the Hamiltonian time rate, equal to the difference between the flux-based and the force-based Rayleigh dissipation functions. In fact, the Fokker–Planck equation can be interpreted as the Hamilton–Jacobi equation resulting from such minumum principle. At steady state, the Hamiltonian time rate is maximized, leading to a minimum resistance principle. In the unsteady case, we consider the relaxation to equilibrium of harmonic oscillators and the motion of a Brownian particle in shear flow, obtaining results that coincide with the solution of the Fokker–Planck and the Langevin equations.

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Notes

  1. Clearly, this is true only when the system is close to its equilibrium configuration, since, in general, phenomenological coefficients are far from being constant; for example, in the case of heat conduction, thermal conductivity, relating heat current and temperature gradient, should vary with temperature as \(T^{-2}\), which is very unlikely, and so the principle of minimum entropy production should be taken with great caution.

  2. Denoting \( \epsilon = t / N \), \( t_i = i \epsilon \) and \( \mathbf x _i = \mathbf x (t_i) \), the path integral is defined as the following functional integral:

    A rigorous and complete treatment on functional integration applied on diffusion processes can be found in Graham [26].

  3. This can also be proved explicitly, multiplying Eq. (23) by \(\dot{\mathbf{y}}\) and considering that \( \frac{d}{dt} = \frac{\partial }{\partial t} + \dot{\mathbf{y}} \cdot \mathbf{\nabla }\), obtaining \( d\mathcal H /dt = 0 \), i.e. \(\mathcal H \) is constant, along the minimum path.

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  1. Department of Civil and Industrial Engineering, Laboratory of Reactive Multiphase Flows, Università di Pisa, 56126, Pisa, Italy

    Roberto Mauri

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  1. Roberto Mauri
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Mauri, R. The Principle of Minimal Resistance in Non-equilibrium Thermodynamics. Found Phys 46, 393–408 (2016). https://doi.org/10.1007/s10701-015-9969-3

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