# On thermal diffusion and gauge transformations for thermodynamic fluxes and forces

## TL;DR

This paper analyzes molecular diffusion in dilute liquids under temperature gradients, highlighting gauge invariance issues in thermodynamic equations and proposing a first-principles approach to thermal diffusion, while clarifying misconceptions about barodiffusion.

## Contribution

It reveals the gauge transformation effects on thermodynamic flux coefficients and suggests a new method for evaluating thermal diffusion constants from entropy considerations.

## Key findings

- Gauge transformations affect cross-effect coefficients in flux equations.
- Thermal diffusion constants can be approached from entropy balance considerations.
- Barodiffusion driven solely by pressure gradients is theoretically impossible in dilute solutions.

## Abstract

We discuss the molecular diffusion transport in infinitely dilute liquid solutions under non-isothermal conditions. This discussion is motivated by an occurring misinterpretation of thermodynamic transport equations written in terms of chemical potential in the presence of temperature gradient. The transport equations contain the contributions owned by a gauge transformation related to the fact that chemical potential is determined up to the summand of form (AT+B) with arbitrary constants A and B, where constant A is owned by the entropy invariance with respect to shifts by a constant value and B is owned by the potential energy invariance with respect to shifts by a constant value. The coefficients of the cross-effect terms in thermodynamic fluxes are contributed by this gauge transformation and, generally, are not the actual cross-effect physical transport coefficients. Our treatment is based on consideration of the entropy balance and suggests a promising hint for attempts of evaluation of the thermal diffusion constant from the first principles. We also discuss the impossibility of the "barodiffusion" for dilute solutions, understood in a sense of diffusion flux driven by the pressure gradient itself. When one speaks of "barodiffusion" terms in literature, these terms typically represent the drift in external potential force field (e.g., electric or gravitational fields), where in the final equations the specific force on molecules is substituted with an expression with the hydrostatic pressure gradient this external force field produces. Obviously, the interpretation of the latter as barodiffusion is fragile and may hinder the accounting for the diffusion fluxes produced by the pressure gradient itself.

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Source: https://tomesphere.com/paper/1705.03690