Schrödinger equations and fluctuation theorems for collisionless plasma systems
Abstract
The fluctuation theorem and detailed fluctuation theorem are formulated for classical systems whose governing equations can be written in Schrödinger-type equations and which possess either a unitary or an antiunitary time-reversal operator. The initial state vector is treated as a random variable drawn from a time-reversal-symmetric probability distribution, and a stochastic relative entropy defined from its probability density is used to formulate these theorems. The framework is applied to two collisionless plasma systems: the linear Vlasov-Poisson and linear gyrokinetic systems. For the linear Vlasov-Poisson system, the governing equations are recast into Schrödinger form, and Hamiltonian eigenvectors corresponding to Case-Van Kampen modes are derived to construct explicit solutions. The stochastic relative entropy is interpreted as entropy generation associated with Landau damping, in which electric-field energy is transferred from the lowest Hermite state to higher-order Hermite states acting as thermal reservoirs. For a specific class of initial distributions, a new analytical expression for the probability density function of the stochastic relative entropy is derived and validated numerically. For the linear gyrokinetic system in a uniform magnetic field, the governing equations are likewise transformed into Schrödinger form, and the corresponding time-reversal operators are identified. The state-vector space is constructed as a tensor product of species, perpendicular-velocity, and parallel-velocity spaces. The resulting state vectors decompose into two orthogonal components: one coupled to electromagnetic fluctuations and the other corresponding to ballistic modes. These results establish a nonequilibrium statistical-mechanical framework for collisionless plasma dynamics and provide useful examples for future quantum-computing applications to plasma simulations.
Versions
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BibTeX
@misc{sugama2026schrödinger,
title = {{Schrödinger equations and fluctuation theorems for collisionless plasma systems}},
author = {Hideo Sugama},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.17656},
eprint = {2606.17656},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.17656},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Schrödinger equations and fluctuation theorems for collisionless plasma systems
AU - Hideo Sugama
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2606.17656
UR - https://arxiv.org/abs/2606.17656
AB - The fluctuation theorem and detailed fluctuation theorem are formulated for classical systems whose governing equations can be written in Schrödinger-type equations and which possess either a unitary or an antiunitary time-reversal operator. The initial state vector is treated as a random variable drawn from a time-reversal-symmetric probability distribution, and a stochastic relative entropy defined from its probability density is used to formulate these theorems. The framework is applied to two collisionless plasma systems: the linear Vlasov-Poisson and linear gyrokinetic systems. For the linear Vlasov-Poisson system, the governing equations are recast into Schrödinger form, and Hamiltonian eigenvectors corresponding to Case-Van Kampen modes are derived to construct explicit solutions. The stochastic relative entropy is interpreted as entropy generation associated with Landau damping, in which electric-field energy is transferred from the lowest Hermite state to higher-order Hermite states acting as thermal reservoirs. For a specific class of initial distributions, a new analytical expression for the probability density function of the stochastic relative entropy is derived and validated numerically. For the linear gyrokinetic system in a uniform magnetic field, the governing equations are likewise transformed into Schrödinger form, and the corresponding time-reversal operators are identified. The state-vector space is constructed as a tensor product of species, perpendicular-velocity, and parallel-velocity spaces. The resulting state vectors decompose into two orthogonal components: one coupled to electromagnetic fluctuations and the other corresponding to ballistic modes. These results establish a nonequilibrium statistical-mechanical framework for collisionless plasma dynamics and provide useful examples for future quantum-computing applications to plasma simulations.
ER -