Effective thermodynamics of two interacting underdamped Brownian particles

Starting from the stochastic thermodynamics description of two coupled underdamped Brownian particles, we showcase and compare three different coarse-graining schemes leading to an effective thermodynamic description for the first of the two particles: marginalization over one particle, bipartite structure with information flows, and the Hamiltonian of mean force formalism. In the limit of time-scale separation where the second particle with a fast relaxation time scale locally equilibrates with respect to the coordinates of the first slowly relaxing particle, the effective thermodynamics resulting from the first and third approach are shown to capture the full thermodynamics and to coincide with each other. In the bipartite approach, the slow part does not, in general, allow for an exact thermodynamic description as the entropic exchange between the particles is ignored. Physically, the second particle effectively becomes part of the heat reservoir. In the limit where the second particle becomes heavy and thus deterministic, the effective thermodynamics of the first two coarse-graining methods coincide with the full one. The Hamiltonian of mean force formalism, however, is shown to be incompatible with that limit. Physically, the second particle becomes a work source. These theoretical results are illustrated using an exactly solvable harmonic model.

Figure 1

On the left, schematics of the two underdamped and via $V_{12}^{\text{int}}$ interacting particles 1 and 2 that are in contact with heat reservoirs at inverse temperatures ${\beta }_1$ and ${\beta }_2$, respectively, are illustrated. It is furthermore assumed that both particles are subjected to nonconservative forces ${\mathit{g}}_i\left({\mathit{x}}_i\right)$. The right depicts the coarse-grained description of solely the first particle in the presence of an additional nonconservative force ${\mathit{g}}^{\left(1\right)}\left({\mathit{\Gamma }}_1\right)$ (green, dashed vector) that encodes the interaction with the second particle.

Figure 2

Difference between the full $\dot{Q}$ and effective heat current ${\dot{Q}}^{\left(1\right)}$ in (a) and between the scaled full $\beta \dot{\mathrm{\Sigma }}$ and scaled effective entropy production rate $\beta {\dot{\mathrm{\Sigma }}}^{\left(1\right)}$ in (b) as a function of time $t$. In these two figures, the parameter sets $a, b$, and $c$ correspond to the solid blue line with circle markers, the solid green line with square markers, and the solid orange line with diamond markers, respectively. Moreover, the effective quantities for the parameters $c$ and those based on the HMF are overlaid in panels (a) and (b) and correspond to the red dashed line and the black pentagon markers, respectively. The information flow ${\dot{I}}^{(2\to 1)}$ as a function of time and for the parameter sets $a$ (solid blue line with circle markers), $b$ (solid green line with square markers), and $c$ (solid orange line with diamond markers) is depicted in panel (c).

Figure 3

Variance ${\mathit{{\rm Y}}}_{11}$ in (a) and ${\mathit{{\rm Y}}}_{33}$ in (b) of the positional degrees of freedom $x_1$ and $x_2$, respectively, as a function of time $t$ and for the different parameter sets $a$ (solid blue line with circle markers), $b$ (solid green line with square markers), and $c$ (solid orange line with diamond markers).

Figure 4

Difference between the full $\dot{Q}$ and effective heat current ${\dot{Q}}^{\left(1\right)}$ in (a) and between the full $\dot{\mathrm{\Sigma }}$ and effective entropy production rate ${\dot{\mathrm{\Sigma }}}^{\left(1\right)}$ in (b) as a function of time $t$. In panel (a) [(b)], the parameter sets $a, b$, and $c$ correspond to the upper [lower] blue intermediate green and lower [upper] orange line, respectively. Moreover, the difference between the full $\dot{Q} \left[\dot{\mathrm{\Sigma }}\right]$ and the naive definition of the effective heat current [entropy production rate] ${\dot{q}}^{\left(1\right)} \left[{\dot{\sigma }}^{\left(1\right)}\right]$ is overlaid in panel (a) [(b)]. The information flow ${\dot{I}}^{(2\to 1)}$ as a function of time and for the parameter sets $a$ [solid blue line with circle markers], $b$ [solid green line with square markers], and $c$ [solid orange line with diamond markers] is depicted in panel (c).