Classical statistical mechanics/thermodynamics is sometimes portrayed as an approximate approach to describing the classical mechanical motion of very large numbers of particles. Equilibrium statistical mechanics/thermodynamics, however, is a special situation in which thermodynamic averages such as temperature, pressure etc do not change in time. We argue in this note that this is due to constraints imposed on such equilibrium scenarios namely time reversal reaction balance. It is this (approximate) constraint which yields the Maxwell-Boltzmann distribution which in turn defines average values of classical values e.g. pressure resulting from an average of bvv (b=constant) and temperature being related to average kinetic energy. Interestingly, average values such as average energy and pressure may yield a classical mechanical relationship for adiabatic processes, otherwise the more complicated dE=TdS - PdV applies to these averages. (To be more accurate one may use the constraints of microcanonical theory.) Quantum mechanics is also a statistical theory because a single particle with a fixed momentum is characterized by a physical wavelength region and a probability distribution exp(ipx). If interactions occur with potential regions a superposition of these distributions (i.e. a momentum superposition) occurs i.e. W(x)= Sum over p a(p)exp(ipx). This represents one constraint in quantum mechanics), but there is a second, namely that averages for a bound system match classical values. Thus, like classical statistical mechanics one may imagine a system characterized by a parameter say L (box length) which changes such that the particle stays in the same quantum state (i.e. an adiabatic change). Given that bound quantum mechanics is constrained to behave on average as a classical system it is not surprising that a dE=PdV which is essentially a Newton’s second law statement dE=Fdx and is not necessarily linked to a usual statistical system. If a quantum state has its parameter changed nonadiabatically it should be able to have it change into a superposition of new quantum states (new L) which is very different from the classical statistical mechanical scenario. Furthermore the ideal gas relationship PV=NRT and E=3/2NRT (3 degrees of freedom) or PV=(⅔)E matches a classical mechanical result because a particle with p may bounce from a wall yielding an impulse. In this note we try to examine some of these underlying constraints and argue that even dE=PdV for average E and P values (i.e. no entropy change) one must be aware of the specific constraints involved. In other words because dE=PdV for both adiabatic classical statistical mechanical systems and a quantum eigenstate, these do not necessarily represent the same statistical type of physics. We further argue that quantum mechanical bound states are constrained to have some average and rms values such as KE(x) and p rms behave as classical values and so these should not be taken to represent “intrinsic” quantum behaviour while average spatial density on the other hand does. (It is linked to the constraint of wavelength.)Classical statistical mechanics/thermodynamics is sometimes portrayed as an approximate approach to describing the classical mechanical motion of very large numbers of particles. Equilibrium statistical mechanics/thermodynamics, however, is a special situation in which thermodynamic averages such as temperature, pressure etc do not change in time. We argue in this note that this is due to constraints imposed on such equilibrium scenarios namely time reversal reaction balance. It is this (approximate) constraint which yields the Maxwell-Boltzmann distribution which in turn defines average values of classical values e.g. pressure resulting from an average of bvv (b=constant) and temperature being related to average kinetic energy. Interestingly, average values such as average energy and pressure may yield a classical mechanical relationship for adiabatic processes, otherwise the more complicated dE=TdS - PdV applies to these averages. (To be more accurate one may use the constraints of microcanonical theory.) Quantum mechanics is also a statistical theory because a single particle with a fixed momentum is characterized by a physical wavelength region and a probability distribution exp(ipx). If interactions occur with potential regions a superposition of these distributions (i.e. a momentum superposition) occurs i.e. W(x)= Sum over p a(p)exp(ipx). This represents one constraint in quantum mechanics), but there is a second, namely that averages for a bound system match classical values. Thus, like classical statistical mechanics one may imagine a system characterized by a parameter say L (box length) which changes such that the particle stays in the same quantum state (i.e. an adiabatic change). Given that bound quantum mechanics is constrained to behave on average as a classical system it is not surprising that a dE=PdV which is essentially a Newton’s second law statement dE=Fdx and is not necessarily linked to a usual statistical system. If a quantum state has its parameter changed nonadiabatically it should be able to have it change into a superposition of new quantum states (new L) which is very different from the classical statistical mechanical scenario. Furthermore the ideal gas relationship PV=NRT and E=3/2NRT (3 degrees of freedom) or PV=(⅔)E matches a classical mechanical result because a particle with p may bounce from a wall yielding an impulse. In this note we try to examine some of these underlying constraints and argue that even dE=PdV for average E and P values (i.e. no entropy change) one must be aware of the specific constraints involved. In other words because dE=PdV for both adiabatic classical statistical mechanical systems and a quantum eigenstate, these do not necessarily represent the same statistical type of physics. We further argue that quantum mechanical bound states are constrained to have some average and rms values such as KE(x) and p rms behave as classical values and so these should not be taken to represent “intrinsic” quantum behaviour while average spatial density on the other hand does. (It is linked to the constraint of wavelength.)

Oct 22, 20210 citations

Regularized Boltzmann entropy determines macroscopic adiabatic accessibility

Hiroyasu Tajima, Eyuri Wakakuwa

How the thermodynamic entropy $S_{TD}$ is related to the Boltzmann entropy $S_{B}$ has been one of the central issues since the beginning of statistical mechanics. Today, it is believed that the thermodynamic entropy $S_{TD}$ is equal to a function $\tilde{S}_{B}$ that is defined by regularizing the Boltzman entropy in order to ensure extensivity. However, it is not known whether $\tilde{S}_{B}$ completely determines the possibility of a macroscopic adiabatic transformation in the same way as $S_{TD}$ does. In this paper, by formulating the possibility of a macroscopic adiabatic transformations in terms of "coarse-graining" of quantum operations, we prove that $\tilde{S}_{B}$ provides a necessary and sufficient condition for the possibility of a macroscopic adiabatic transformation.How the thermodynamic entropy $S_{TD}$ is related to the Boltzmann entropy $S_{B}$ has been one of the central issues since the beginning of statistical mechanics. Today, it is believed that the thermodynamic entropy $S_{TD}$ is equal to a function $\tilde{S}_{B}$ that is defined by regularizing the Boltzman entropy in order to ensure extensivity. However, it is not known whether $\tilde{S}_{B}$ completely determines the possibility of a macroscopic adiabatic transformation in the same way as $S_{TD}$ does. In this paper, by formulating the possibility of a macroscopic adiabatic transformations in terms of "coarse-graining" of quantum operations, we prove that $\tilde{S}_{B}$ provides a necessary and sufficient condition for the possibility of a macroscopic adiabatic transformation.

arXiv:1601.00487Jan 4, 20163 citations

On the Relation Between the Second Law of Thermodynamics and Classical and Quantum Mechanics

Barbara Drossel

arXiv:1408.6358The frontiers collectionJan 1, 201527 citations

Nonlocal contributions to ergotropy: A thermodynamic perspective

B. Vigneshwar, R. Sankaranarayanan

Nonlocality is a defining feature of quantum mechanics and has long served as a key indicator of quantum resources since the formulation of Bell's inequalities. Identifying the contribution of nonlocality to extractable work remains a central problem in quantum thermodynamics. We address this by introducing a quantifier of nonlocal contributions to extractable work in bipartite systems. It is shown that closed form expressions can be calculated for our quantity in terms of the Schmidt coefficients. Further for strictly non-interacting Hamiltonian, the direct relationship between ergotropy and correlations is established. Our results reveal that nonlocal resources invariably enhance extractable work under non-interacting Hamiltonians, while in the presence of interactions, their contribution can either increase or diminish depending on the structure of the state and the Hamiltonian.Nonlocality is a defining feature of quantum mechanics and has long served as a key indicator of quantum resources since the formulation of Bell's inequalities. Identifying the contribution of nonlocality to extractable work remains a central problem in quantum thermodynamics. We address this by introducing a quantifier of nonlocal contributions to extractable work in bipartite systems. It is shown that closed form expressions can be calculated for our quantity in terms of the Schmidt coefficients. Further for strictly non-interacting Hamiltonian, the direct relationship between ergotropy and correlations is established. Our results reveal that nonlocal resources invariably enhance extractable work under non-interacting Hamiltonians, while in the presence of interactions, their contribution can either increase or diminish depending on the structure of the state and the Hamiltonian.

quant-pharXiv:2512.14497Dec 17, 20250 citations

Adiabatic results in quantum theory and quantum-classical coupling

Folkmar A. Bornemann

Lecture notes in mathematicsJan 1, 19980 citations

How the thermodynamic entropy

S_{TD}

is related to the Boltzmann entropy

S_{B}

has been one of the central issues since the beginning of statistical mechanics. Today, it is believed that the thermodynamic entropy

S_{TD}

is equal to a function

\tilde{S}_{B}

that is defined by regularizing the Boltzman entropy in order to ensure extensivity. However, it is not known whether

\tilde{S}_{B}

completely determines the possibility of a macroscopic adiabatic transformation in the same way as

S_{TD}

does. In this paper, by formulating the possibility of a macroscopic adiabatic transformations in terms of "coarse-graining" of quantum operations, we prove that

\tilde{S}_{B}

provides a necessary and sufficient condition for the possibility of a macroscopic adiabatic transformation.