# A Thermal Force Drifting Particles Along a Temperature Gradient

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1 Polito - Politecnico di Torino Torino

Abstract : In 1972, V. Gallina and M. Omini of the Polytechnic of Turin proposed a phenomenological model for the thermal diffusion in liquid metals, explaining the isotope separation as provoked by a thermal force which is arising when a temperature gradient is established in the material. Here, we discuss this thermal force and its statistical origin from the bulk. We will see that it can be considered as a force of the form F = − S grad T, that is as a thermal-entropic force obtained from the derivative of the Helmholtz free energy with respect to the volume. Introduction. With the works of Ludwig Boltzmann, physics and thermodynamics started recognizing the stochastic and probabilistic aspect of natural processes. Besides introducing a fundamental equation widely used for thermal and charge transport 1-5, Boltzmann linked the second law of thermodynamics to the disorder of thermodynamic systems, proposing a fundamental relation between entropy S and probability through the formula S = kB ln W kB is the Boltzmann-s constant and W is for Wahrscheinlichkeit, that is - probability - in German. The Boltzmann formulation of entropy was fundamental for the - doctrine of energy and entropy - that rules the thermodynamic world 6, a doctrine where the energy is deterministic and the entropy is favouring randomness. In this framework of natural systems governed by energy and entropy, forces can arise from the Helmholtz free energy A = U – T S , where U is the internal energy, T the temperature and S the entropy. Such as the pressure, the mechanical force divided by a surface, is coming from the derivative of internal energy U with respect to the volume, the same derivative of the product T S is giving rise to the thermal and entropic forces. These are forces which are coming from the statistical nature of the system, rather than from a particular microscopic interaction existing in it 6. Entropic forces had been proposed and used in 7,8 for the Brownian motion and for the elasticity of polymers. For the Brownian motion, the force is in the form of a diffusional driving force or radial force, which has a mean value = T, where r is a radial distance 9. We can understand this mean value considering its dimensionality: the force is an energy-here given by the product TS-divided by a length. However, besides the entropic force which is coming from a local variation or gradient of entropy, we can have also a force in which it is appearing as the gradient of temperature T multiplied by an entropy, that is F = − S grad T. A thermal force of this kind is used for the magnetic flux structures in superconductors 10,11. In these references, this force is also considered as an entropic force.

Keywords : thermal gradient thermal transport thermal forces entropic forces

Author: ** Amelia Carolina Sparavigna - **

Source: https://hal.archives-ouvertes.fr/