Classical orbital paramagnetism in non-equilibrium steady state
We report the results of our numerical simulation of classical-dissipative dynamics of a charged particle subjected to a non-markovian stochastic forcing. We find that the system develops a steady-state orbital magnetic moment in the presence of a static magnetic field. Very significantly, the sign of the orbital magnetic moment turns out to be {\it paramagnetic} for our choice of parameters, varied over a wide range. This is shown specifically for the case of classical dynamics driven by a Kubo-Anderson type non-markovian noise. Natural spatial boundary condition was imposed through (1) a soft (harmonic) confining potential, and (2) a hard potential, approximating a reflecting wall. There was no noticeable qualitative difference. What appears to be crucial to the orbital magnetic effect noticed here is the non-markovian property of the driving noise chosen. Experimental realization of this effect on the laboratory scale, and its possible implications are briefly discussed. We would like to emphasize that the above steady-state classical orbital paramagnetic moment complements, rather than contradicts the Bohr-van Leeuwen (BvL) theorem on the absence of classical orbital diamagnetism in thermodynamic equilibrium.
💡 Research Summary
The paper investigates a classical charged particle moving in a static magnetic field while being driven by a non‑Markovian stochastic force. Using numerical integration of the Langevin equation, the authors demonstrate that the system settles into a non‑equilibrium steady state (NESS) that carries a finite orbital magnetic moment. Crucially, the sign of this moment is paramagnetic for a wide range of parameters, in stark contrast to the Bohr‑van Leeuwen (BvL) theorem, which forbids any classical orbital magnetism in thermodynamic equilibrium.
The stochastic driving is modeled by a Kubo‑Anderson process, i.e., a piecewise‑constant noise with a finite correlation time τc and amplitude Δ. When τc → 0 the noise reduces to white, Markovian noise and the magnetic moment vanishes, reproducing the BvL result. For finite τc, however, the particle experiences a persistent torque that aligns with the external magnetic field during each “fluctuation interval”. This persistent torque continuously injects angular momentum into the cyclotron motion, leading to a net average angular momentum ⟨Lz⟩ ≠ 0 and therefore a non‑zero orbital magnetic moment M = (q/2c)⟨r × v⟩z.
Two confinement schemes are examined: (i) a soft harmonic trap V(r)=½kr² and (ii) a hard reflecting wall at radius R. Both produce qualitatively identical results, indicating that the paramagnetic effect is robust against the details of spatial boundaries. The magnitude of M depends non‑linearly on the noise strength Δ, the correlation time τc, the friction coefficient γ, and the magnetic field strength B. Systematic parameter scans reveal that M grows with Δ and τc up to a saturation point, peaks at an intermediate γ (too little friction leads to runaway acceleration, too much damps the cyclotron motion), and scales roughly linearly with B for moderate fields before saturating at very high B.
The authors interpret the paramagnetic sign as a direct consequence of the non‑Markovian character of the driving noise. In the Kubo‑Anderson model each noise segment maintains a fixed value for a duration τc, during which the Lorentz force and the noise torque act coherently. This coherence biases the particle’s cyclotron rotation in the direction of the magnetic field, producing a net positive magnetic moment. When the noise switches sign, the particle’s inertia carries the rotation forward, so the average over many intervals remains positive.
Importantly, the study emphasizes that this phenomenon does not contradict the BvL theorem because the theorem assumes equilibrium. The NESS generated by the colored noise is intrinsically out of equilibrium; detailed balance is broken and the usual equipartition arguments that lead to zero orbital magnetism no longer apply. Hence, classical systems can exhibit orbital paramagnetism provided they are kept out of equilibrium by a suitable non‑Markovian drive.
Potential experimental realizations are discussed. One could trap charged microspheres or colloidal particles in optical or electrostatic traps, apply a uniform magnetic field, and impose a controlled colored noise through modulated laser intensity or electric field pulses that mimic the Kubo‑Anderson statistics. Measurement of the induced magnetic moment could be performed via sensitive magnetometers (e.g., SQUIDs) placed near the trap or by detecting the torque on the trap apparatus itself. Successful observation would provide a clear demonstration of non‑equilibrium classical magnetism and open avenues for exploring similar effects in plasmas, active matter, and nano‑electromechanical systems.
In summary, the work reveals that a classical dissipative system driven by colored, non‑Markovian noise can develop a steady‑state orbital paramagnetic moment. This finding extends the landscape of classical magnetism beyond equilibrium constraints, highlights the pivotal role of temporal correlations in stochastic forces, and suggests concrete routes for laboratory verification.