# Different measures for characterizing the motion of molecules along a temperature gradient

## TL;DR

This paper analyzes the motion of Brownian particles in temperature gradients, deriving time-dependent probability distributions and examining how different measures can lead to ambiguous interpretations of thermophoretic behavior.

## Contribution

It introduces a time-dependent analysis of particle motion in temperature gradients, highlighting ambiguities in characterizing thermophoretic responses using traditional measures.

## Key findings

- Mean displacement and bias can have opposite signs to the Soret coefficient.
- Time-dependent PDFs reveal complex thermophoretic dynamics.
- Different measures influence the interpretation of thermophoretic behavior.

## Abstract

We study the motion of a Brownian particle in a medium with inhomogeneous temperature. In the overdamped regime of low Reynolds numbers, the probability distribution function (PDF) of the particle is obtained from the van Kampen diffusion equation [J. Phys. Chem. Solids {\bf 49}, 673 (1988)]. The thermophoretic behavior is commonly described by the Soret coefficient - a parameter which can be calculated from the steady-state PDF. Motivated by recent advances in experimental methods for observing and analyzing single nano-particle trajectories, we here consider the time-dependent van Kampen equation from which the temporal evolution of the PDF of individual particles can be derived. We analytically calculate the PDF describing dynamics driven by a generalized thermophoretic force. Single particles statistics is characterized by measures like the mean displacement (drift) and the probability difference between moving along and against the temperature gradient (bias). We demonstrate that these quantities do not necessarily have the same sign as the Soret coefficient, which causes ambiguity in the distinction between thermophilic and thermophobic response (i.e., migration in and against the direction of the temperature gradient). The different factors determining the thermophoretic response and their influence on each measure are discussed.