Nonequilibrium driven by an external torque in the presence of a magnetic field

We investigate a two-dimensional motion of a colloid in a harmonic trap driven out of equilibrium by an external nonconservative force producing a torque in the presence of a uniform magnetic field applied perpendicular to the plane of motion. We find a circulating steady-state current diagnostic to nonequilibrium. Unlikely in the overdamped limit, inertial motion requires a sufficient central force to reach steady state. The magnetic field can enhance or depress central force depending on its direction. We find that steady state exists only for a proper range of parameters such as mass, viscosity coefficient, stiffness of the harmonic potential, and the magnetic field. We rigorously derive the existence condition for the steady state. We examine the combined influence of nonconservative force and magnetic field on nonequilibrium characteristics. We find non-Boltzmann steady-state probability density function and circulating probability current. We show that nonnegative entropy production is composed of usual heat dissipation and unconventional contribution from velocity-dependence of the Lorentz force. We derive the full list of correlation functions, including position-velocity correlation function originated from nonequilibrium circulation. We finally give rigorous expression for the violation of fluctuation-dissipation relation. We verify our analytical results by using the Monte Carlo simulation.

Figure 1

A circular orbit and involved forces. The dissipative force is $-\gamma \vec{v}$. The figure is drawn for $a,b>0$. The harmonic force and magnetic force for $ab>0$ ($ab<0$) are in the same (opposite) direction so that they strengthen (weaken) centripetal force.

Figure 2

The existence region for steady state in a parameter space for $k$ and $a>0$. The stable region is above the boundary line. Boundaries are drawn for $b=-3\gamma ,0,3\gamma$. The larger $b$ and the smaller $a$, the more widened the stable region.

Figure 3

Contour plots of the probability density function and velocity fields for (a) isotropic and (b) nonisotropic cases. The vertical and horizontal axes represent $x$ and $y$ in unit of $\gamma /\sqrt{m{k}_{B}T}$. Velocity field lines are shown to circulate around the contour lines for both cases.

Figure 4

The plot of $C_{x{v}_y}\left(\tau \right)$ for various $a,b>0$ with fixed $k=3{\gamma }^2/m$. There is a nonvanishing correlation for $b\ne 0$ and $a=0$, which is distinguishable from normal equilibrium. The amplitude of the correlation decreases as $b$ increases. Four lines represent analytic results. We verify the analytic results with Monte Carlo simulation. Dots, squares, triangular, and cross represent the simulation result.

Figure 5

The Violation of the FDR in position basis: $V_{y{v}_x}\left(\tau \right)$ for $k={\gamma }^2/m$ and $a=2{\gamma }^2/m$. The fluctuation from the guide line is more amplified for smaller $b$ (smaller $\mathrm{\Omega }$). Dots and squares represent the points obtained from the simulation with ${10}^4$ ensembles.