System-Bath Approach to Rotating Brownian Motion

TL;DR

This work analyzes Brownian motion coupled to a rotating thermal bath using a solvable Caldeira–Leggett model and reveals that rotation induces long-range correlated noise and a stationary state that is non-Gibbsian at finite coupling. In the weak-coupling limit, the Brownian particle approaches a rotating Gibbs distribution with temperature $T$ and effective rotation $\Omega_{\rm eff}$; with magnetic fields, however, the stationary state becomes non-Gibbsian and can be described by effective parameters $T_{\rm eff}$ and $\Omega_{\rm eff}$ that depend on the field and spectrum of the bath. The paper shows that a free energy exists for rotation-symmetric potentials, implying no work can be extracted through cyclic, slow processes in that case, while non-rotation-symmetric potentials permit finite work extraction. Applications to sedimentation demonstrate centrifugal stability is preserved by finite friction, and the authors propose experimental tests with optically trapped colloids in a rotating bath to observe the predicted effects. Overall, the study provides a microscopic foundation for rotating Brownian dynamics, clarifies the thermodynamics under rotation, and highlights conditions under which work can be extracted in rotating systems.

Abstract

Rotating equilibrated systems are widespread, but relatively little attention has been devoted to studying them from the first principles of statistical mechanics. We fill this gap by studying a Brownian particle coupled with a thermal bath made of rotating harmonic oscillators. We show that the Langevin equation that describes the dynamics of the Brownian particle contains (due to rotation) long-range correlated noise. In contrast to the usual situation of (non-rotating) equilibration, the rotating Gibbs distribution is recovered only for a weak coupling with the bath. In the presence of a uniform magnetic field, the stationary state is not Gibbsian, even under weak coupling. In this context, we clarify the applicability of the Bohr-van Leeuwen theorem to classical systems in rotating equilibrium, as well as the concept of work done by a changing magnetic field. We show that the Brownian particle under a rotationally symmetric potential reaches a stationary state that behaves as an effective equilibrium, characterized by a free energy. As a result, no work can be extracted via cyclic processes that respect the rotation symmetry. However, if the external potential exhibits asymmetry, then work extraction via slow cyclic processes is possible. This is illustrated by a general scenario involving a slow rotation of a non-rotation-symmetric potential. We study sedimentation equilibrium and show that centrifugal instability is prevented by a finite friction.

Figures (9)

• Figure 1: Illustration of the system+bath. The system is a Brownian charge coupled to harmonic oscillators (thermal bath) with strength $c_k$. Each oscillator $k$ with frequency $\omega_k$ rotates with its own frequency $\Omega_k$; see \ref{['stability']}.
• Figure 2: Visualisation of the angular velocity spectrum of the oscillators \ref{['omega-tanh']} with $a=0.8$ (solid lines). For large $\Omega_0$, this approaches the linear limit $a\,\omega$ in \ref{['linear']}.
• Figure 3: Plot of function $f(x)$ defined in \ref{['f-def']}.
• Figure 4: Given (\ref{['integrals-1']}--\ref{['integrals-3']}), we derive the effective temperature $T_{\rm eff}$ (left figure) and effective angular velocity $\Omega_{\rm eff}$ (right figure) for the Brownian oscillator using the formulas in Appendix \ref{['app-harmonic-gibbs']}. The dotted lines represent numerical evaluations with the nonlinear angular velocity spectrum \ref{['omega-tanh']} with parameters $\Omega_0$ and $a$, while the dashed line represents the analytically solvable case of linear spectrum \ref{['linear']} with $a=0.8$. We note that for $\Omega_0\gg\omega_0$, the nonlinear spectrum approaches the linear case, as also seen from Figure \ref{['fig-omegas']} and the plots above. Furthermore, in the weak coupling limit (low friction), the numerical estimates (dotted lines) converge to the corresponding parameters of the thermal bath (dashed line), i.e., $T^{\rm eff}\approx T$ and $\Omega^{\rm eff} \approx \Omega$, where $T$ corresponds to $1/\beta$ temperature of the thermal bath, defined in \ref{['initial_state_4']}. The ratio $\Omega^{\rm eff}/\Omega(\omega_0)$ has non-monotonic dependence on $\Omega_0$. In the inset, we fix $\gamma=\omega_0$ and plot the dependence on $\Omega_0$. The dashed line in the inset corresponds to the theoretical value for $\Omega_0 \gg \omega_0$. It is seen that $\Omega_{\rm eff}\leq \Omega(\omega_0)$ and $T_{\rm eff}\geq T$, while $\Omega_{\rm eff}\to \Omega(\omega_0)$ and $T_{\rm eff}\to T$ for $\gamma\to 0$.
• Figure 5: Sedimentation of particles of different masses coupled to a rotating thermal bath. Dependence of (scaled) mean squared displacement on the (scaled) mass of the particle; see (\ref{['obukh']}, \ref{['omega-tanh']}).