Partition of kinetic energy and magnetic moment in dissipative diamagnetism

Independent-oscillator model and the quantum Langevin equation for an oscillator: a review

“…This feature has been analyzed in detail in [46]. It may be remarked that when the particle is charged and there is a magnetic field acting on it, the number of peaks increases to two or three depending on the embedding spatial dimension of the Hamiltonian [42,46]. Combining the two cases discussed above, we may obtain the average of the total energy of the system in equilibrium, as…”

“…Although it may come as a surprise that thermodynamics can be consistently formulated for a single particle, which is far from the thermodynamic limit that one invokes in traditional thermodynamic studies, it turns out that not only does a thermal description follow naturally from the framework of classical and quantum Brownian motion, it is also consistent with all the laws of thermodynamics, including the third law (see, for example, [26][27][28]35]). We will contrast the expressions of average energy between classical and quantum Brownian oscillators, which will lead us to the recently-proposed quantum counterpart of energy equipartition theorem [36][37][38][39][40][41][42][43][44][45][46]. As an application of the framework relevant to condensed-matter physics, we shall discuss dissipative diamagnetism [46,47] and emphasize on the role of confining potentials.…”

“…We will contrast the expressions of average energy between classical and quantum Brownian oscillators, which will lead us to the recently-proposed quantum counterpart of energy equipartition theorem [36][37][38][39][40][41][42][43][44][45][46]. As an application of the framework relevant to condensed-matter physics, we shall discuss dissipative diamagnetism [46,47] and emphasize on the role of confining potentials. Following this, we will discuss a situation where the system (the oscillator) and the bath are connected via momentum-dependent couplings [48][49][50][51]; the specific case of a gauge-invariant model of a three-dimensional oscillator coupled to a heat bath via momentum variables in the presence of a vector potential is carefully analyzed [52,53].…”

“…It may be shown that P k (ω) is both positive definite and is normalized, justifying its interpretation as a bona fide probability density (see appendix B). In recent literature, equation ( 63) has been termed as the quantum counterpart of energy equipartition theorem [36][37][38][39][40][41][42][43][44][45][46]81]. The justification behind this terminology can be understood from the high-temperature limit of equation ( 63), for which, upon using lim Let us turn our attention to the potential energy.…”

“…In order to incorporate confinement effects correctly, it is important to first evaluate the integral appearing in equation (121) and then take the limit ω 0 → 0 [47,99]. Performing the integration in equation ( 121), we find [46,47]…”