Effect of a magnetic field on the thermodynamic uncertainty relation

Abstract

The thermodynamic uncertainty relation provides a universal lower bound on the product of entropy production and the fluctuations of any current. While proven for Markov dynamics on a discrete set of states and for overdamped Langevin dynamics, its status for underdamped dynamics is still open. We consider a two-dimensional harmonically confined charged particle in a magnetic field under the action of an external torque. We show analytically that, depending on the sign of the magnetic field, the thermodynamic uncertainty relation does not hold for the currents associated with work and heat. A strong magnetic field can effectively localize the particle with concomitant bounded fluctuations and low dissipation. Numerical results for a three-dimensional variant and for further currents suggest that the existence of such a bound depends crucially on the specific current.

Thermodynamic uncertainty relation for underdamped Langevin systems driven by a velocity-dependent force

“…As already reported in Ref. [26], the origianl TUR is satisfied as Q mag ori,∞ (W mag ) ≥ 2 for Bκ < 0, but does not hold for Bκ > 0. We remark that the original TUR is valid for all parameter regions when we take the Lorentz force as the irreversible one.…”

“…Recently, Chun et al [26] studied the validity of the original TUR for a Brownian particle in a magnetic field. They found that the original TUR on the work current can be broken due to a magnetic field.…”

“…As we are interested in the steady state, we need a condition for guaranteeing a stable solution of this linear system which is given by [26,56]…”

Operationally Accessible Bounds on Fluctuations and Entropy Production in Periodically Driven Systems

“…For steady-state heat engines, the relation shows that an inevitable side-effect of reaching Carnot efficiency at finite power are diverging power fluctuations [29][30][31][32][33]. Refinements and generalizations of the TUR have been found for diffusive dynamics [34][35][36], for data allocated over a finite time [37][38][39][40], for ballistic transport [41], for underdamped Langevin dynamics with and without magnetic fields [42][43][44]. Rather than looking at the fluctuations of currents, it is also possible to constrain the fluctuations of time-symmetric quantities like residence times or activity [45][46][47] and the fluctuations of first passage times [48,49].…”

Thermodynamic uncertainty relations in a linear system

“…A connection between discrete and continuous time uncertainty relations is shown in [34]. One can also see similar uncertainty relations in the context of discrete processes [35], multidimensional systems [36], Brownian motion in the tilted periodic potential [37], general Langevin systems [38], molecular motors [39], run and tumble processes [40], biochemical oscillations [41], interacting oscillators [42], effect of magnetic field [43], linear response [44], measurement and feedback control [45], information [46], underdamped Langevin dynamics [47], timedelayed Langevin systems [48], various systems [49], etc.. Recently, Hasegawa et al [50] found an uncertainty relation for the time-asymmetric observable for the system driven by a time-symmetric driving protocol using the steady state fluctuation theorem.…”