Higher-order near-equilibrium fluctuations and dynamical equilibria with several absolute constants of the motion in the Born–Green kinetic theory of thermodynamics

Born and Green's 1948 paper “The kinetic basis of thermodynamics” derives the entropy increase, the approach to equilibrium, and Boltzmann's law for a single weakly perturbed quantum system by expanding the diagonal elements of the density matrix as a series $a(\lambda) = a_0(\lambda) + a_2(\lambda) + a_3(\lambda) + \cdots$ in powers of the perturbation, keeping only the second-order term. The authors leave two extensions unexplored. First, near equilibrium the second-order term $a_2(\lambda)$ vanishes, so the neglected higher-order terms become comparable to it and are expected to appear as continuous random fluctuations about the equilibrium state — suggested as a new route to Brownian movement and the shot effect, with no detailed theory developed. Second, the equilibrium derivation assumes that energy is the only absolute constant of the motion; when further quantities such as total angular momentum or total linear momentum are also conserved, the corresponding dynamical equilibria should depend on these variables, and their treatment under more general conditions is deferred to future work. The underlying physics content — a microscopic theory of equilibrium fluctuations, and equilibria with many conserved quantities — was subsequently developed by other frameworks: the fluctuation-dissipation theorem, projection-operator and quantum-noise theory, and generalized Gibbs ensembles.

Background

By the late 1940s the laws of thermodynamics were usually justified by statistical mechanics through ensembles — hypothetical collections of copies of the system, a step whose physical foundation was widely felt to be obscure — or by quantum arguments that simply postulated transition probabilities between stationary states. In a 1948 paper, Max Born and H. S. Green set out instead to derive the thermodynamics of a single system of any kind directly from quantum mechanics: the increase of entropy in the two fundamental irreversible processes (the conversion of work into heat, and the flow of heat between coupled subsystems), the approach to equilibrium, and Boltzmann's law.

Their machinery runs as follows. The state of the system is described by a density matrix, the quantum operator whose diagonal elements in a chosen basis give the occupation probabilities of the basis states. The basis is taken to be the eigenstates of a complete set of absolute constants of the motion — observables that commute with both the unperturbed Hamiltonian $H$ and the full Hamiltonian $\mathfrak{H} = H + V$ of the weakly perturbed system; each state is labelled by the corresponding eigenvalues $\lambda$. The initial density matrix is assumed diagonal in this basis, with diagonal elements $a_0(\lambda)$. Under the perturbation, states $\lambda$ and $\lambda'$ are connected by second-order transition probabilities $p(\lambda,\lambda') = \frac{2\pi}{\hbar}|\mathfrak{B}(\lambda,\lambda')|^2\, t'\,\delta(\mathfrak{H}-\mathfrak{H}')$ — the golden-rule form (Fermi's golden rule, the standard weak-perturbation formula for quantum transition rates), with $\mathfrak{B}(\lambda,\lambda')$ the matrix element of the perturbation between the two states, $t'$ the elapsed time, and $\delta$ the Dirac delta enforcing energy conservation. The paper's appendix gives an exact formal solution of the equation of motion as an infinite product of exponentials of nested commutator-derived operators, from which the diagonal probabilities expand as

$a(\lambda) \;=\; a_0(\lambda) + a_2(\lambda) + a_3(\lambda) + \cdots,$

where the leading term has the structure $a_2(\lambda) = \sum_{\lambda'} p(\lambda,\lambda')\,\{a_0(\lambda) - a_0(\lambda')\}$. Keeping only this term, the entropy $S = -k \sum_\lambda a(\lambda)\log a(\lambda)$ (with $k$ Boltzmann's constant and an appropriate normalization) can never decrease — a quantum H-theorem, the analogue of Boltzmann's classical theorem that a monotone entropy-like quantity increases until equilibrium is reached. Equilibrium is characterized by the condition $a(\lambda) = a(\lambda')$ for every pair of states connected by a non-vanishing transition probability. For a composite system of weakly coupled subsystems the joint distribution approximately factorizes, the entropy is additive, and if the total energy $H = \sum_i H^{(i)}$ is the only absolute constant of the motion, the equilibrium condition together with additivity forces $\log a$ to be a linear function of energy, giving Boltzmann's law $a(H) = e^{\beta(A-H)}$ with $\beta$ the inverse temperature and $A$ a normalization constant fixed by the total probability.

Two limits of this derivation were pointed out by Born and Green themselves. First, the expansion is truncated at second order. As equilibrium is approached, $a_2(\lambda) \to 0$ because it is proportional to probability differences between connected states, and the discarded higher-order terms $a_3, a_4, \ldots$ become comparable to it; the authors suggest that these terms, producing continuous random fluctuations about the equilibrium state, open a new approach to two by-then-classic noise phenomena — Brownian movement (the irregular thermal motion of small suspended particles) and the shot effect (Schottky noise, the random current fluctuations caused by the discreteness of charge) — but they develop no detailed theory. Second, Boltzmann's law was derived only for the case in which energy is the sole absolute constant of the motion. In systems where further quantities — total angular momentum, total linear momentum, or other additive constants — also commute with both $H$ and $H + V$, the dynamical equilibria may involve distributions depending on these variables, and the paper leaves the treatment of such dynamical equilibria under more general conditions to future work.

For orientation, both questions were later settled in substance by other frameworks: equilibrium fluctuation theory by the fluctuation-dissipation theorem (the relation tying the spectrum of a system's equilibrium fluctuations to its linear-response coefficients) together with projection-operator and quantum-noise methods, and multi-conserved-quantity equilibria by maximum-entropy and generalized Gibbs ensembles (statistical ensembles whose weights are exponentials of linear combinations of all conserved quantities, not of the energy alone). What those frameworks do not provide is the construction inside the Born–Green derivation itself, whose assumptions (an isolated single system, weak perturbation, an initially diagonal density matrix) are more restrictive than the settings in which the modern theories are proved; the unresolved target is precisely the completion of that derivation.

Problem Statement

The 1948 Born–Green kinetic derivation of thermodynamics is carried out only in second order of perturbation theory and only for the case in which the total energy is the single absolute constant of the motion. The source leaves both extensions open; the problem is to complete the derivation along these two directions, which would turn it from a theory of the 1948 special case into a theory of the equilibrium state and of the fluctuations about it for general weakly coupled quantum systems.

(1) Near-equilibrium fluctuations from the higher-order terms. In the expansion $a(\lambda) = a_0(\lambda) + a_2(\lambda) + a_3(\lambda) + \cdots$ of the diagonal state probabilities, the second-order term vanishes at equilibrium, so the higher-order terms become comparable to it there. Develop the theory of this near-equilibrium regime within the Born–Green framework (isolated system, weak perturbation or weak inter-subsystem coupling, density matrix initially diagonal in the basis of the constants of the motion). This part has three sub-questions: (1a) the third- and higher-order terms — including the part of $a_3$ that grows linearly in time — that is, the exact terms together with the leading time dependence they produce; (1b) the statistics, correlation functions, and noise spectra of the resulting fluctuations of $a(\lambda)$ about the equilibrium distribution — that is, of the deviation $\delta a(\lambda,t) := a(\lambda,t) - a_{\rm eq}(\lambda)$ from the equilibrium value $a_{\rm eq}(\lambda)$, with the averages taken over the ensemble fixed by the initial diagonal density matrix $a_0(\lambda)$ (for instance the two-time correlation $\langle \delta a(\lambda,t)\,\delta a(\lambda,0)\rangle$); and (1c) an explicit determination, from those correlation functions and spectra, of whether the theory reproduces Brownian movement and the shot effect or whether the framework's own assumptions are insufficient for genuinely stochastic fluctuations — and in that case which specific feature of the framework's own deterministic evolution prevents the required stochastic structure.

(2) Dynamical equilibria with several absolute constants of the motion. The equilibrium condition $a(\lambda) = a(\lambda')$ for states connected by the perturbation, combined with factorization of the joint distribution of weakly coupled subsystems and additivity of the conserved quantities, was applied only to the energy. Extend it to systems in which additional quantities — total angular momentum, total linear momentum, or any further additive absolute constant of the motion — are also conserved. This part has three sub-questions: (2a) the form of the corresponding dynamical equilibria; (2b) the accompanying H-theorem (monotone increase of the entropy subject to the fixed values of all the constants); and (2c) the reduction of the general equilibrium obtained in (2a) to Boltzmann's law $a(H) = e^{\beta(A-H)}$ when only the energy is conserved, which checks the general result against the energy-only case.

Parts (1) and (2) are two distinct open directions from the same 1948 framework. (1c) is determined from the correlation functions and spectra of (1b), and (2c) checks (2a) against the energy-only case. Known points of contact — fluctuation-dissipation relations for part (1), generalized Gibbs ensembles for part (2) — are relevant background for interpreting results.

References

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