Negative Thermophoretic Force in the Strong Coupling Regime

Negative thermophoresis (a particle moving up the temperature gradient) is a somewhat counterintuitive phenomenon which has thus far eluded a simple thermostatistical description. The purpose of this Letter is to show that a thermodynamic framework based on the formulation of a Hamiltonian of mean force has the descriptive ability to capture this interesting and elusive phenomenon in an unusually elegant and straightforward fashion. We propose a mechanism that describes the advent of a thermophoretic force acting from cold to hot on systems that are strongly coupled to a nonisothermal heat bath. When a system is strongly coupled to the heat bath, the system’s eigenenergies ${\mathcal{E}}_j$ become effectively temperature dependent. This adjustment of the energy levels allows the system to take heat from the environment, $+d⟨{\mathcal{E}}_j⟩$, and return it as work, $-d⟨Td{\mathcal{E}}_j/dT⟩$. This effect can make the temperature dependence of the effective energy profile nonmonotonic. As a result, particles may experience a force in either direction depending on the temperature.

Figure 1

The energy levels ${\mathcal{E}}_j(T)$ determine the probability that state $j$ is occupied. Once level $j$ is occupied, its contribution to the system’s energy is given by ${\mathrm{\Phi }}_j(T)$. While the ${\mathcal{E}}_j(T)$ are monotonically increasing functions of temperature, at low enough temperature the ${\mathrm{\Phi }}_j(T)$ exhibit the opposite behavior.