Heat and work are fundamental concepts for thermodynamical systems. When these are scaled down to the quantum level they require appropriate embeddings. Here we show that the dependence of the particle spectrum on system size giving rise to a formal definition of pressure can, indeed, be correlated with an external mechanical degree of freedom, modelled as a spatial coordinate of a quantum oscillator. Under specific conditions this correlation is reminiscent of that occurring in the classical manometer.
Heat and work are fundamental concepts for thermodynamical systems. When these are scaled down to the quantum level they require appropriate embeddings. Here we show that the dependence of the particle spectrum on system size giving rise to a formal definition of pressure can, indeed, be correlated with an external mechanical degree of freedom, modelled as a spatial coordinate of a quantum oscillator. Under specific conditions this correlation is reminiscent of that occurring in the classical manometer.
Quantum systems in a mechanical embedding, the breathing mode of a small particles, optomechanical system, etc. are far not the full list of examples in which the volume exhibits quantum behavior. Traditional consideration suggests strain in small systems as a result of a collective movement of particles, rather than the dynamics of the volume as an independent variable. The aim of this work is to show that some problem here might be essentially simplified by introducing periodic boundary conditions. At this case, the volume is considered as the independent dynamical variable driven by the internal pressure. For this purpose, the concept of quantum volume based on Schrödinger’s equation in [Formula: see text] manifold is proposed. It is used to explore several 1D model systems: An ensemble of free particles under external pressure, quantum manometer and a quantum breathing mode. In particular, the influence of the pressure of free particle on quantum oscillator is determined. It is shown also that correction to the spectrum of the breathing mode due to internal degrees of freedom is determined by the off-diagonal matrix elements of the quantum stress. The new treatment not using the “force” theorem is proposed for the quantum stress tensor. In the general case of flexible quantum 3D dynamics, quantum deformations of different type might be introduced similarly to monopole mode.
Classical statistical mechanics tries to derive the macroscopic properties of matter starting from the mechanical laws that rule the behaviour of single particles. To describe equilibrium states, only observables accounting for correlations among states are considered, and only systems consisting of a large number of particles are taken into account. The observables are described by continuous functions on phase space, and the states are represented by linear assignments of a number to each observable. Within the framework of the canonical ensemble, one deals with mechanical systems in thermal equilibrium with a thermal reservoir, and the equilibrium state is reached as a result of the interaction with the external world. The external world may be really external, or equally well, the unrecognized internal degrees of freedom, like in the calculation of viscosity or thermal conductivity. If really only the external world is the cause we may expect some surface dependence while an internal unrecognized degree of freedom would have a volume effect, unless the interactions are long range. The microcanonical ensemble is instead introduced to study isolated mechanical systems, and the equilibrium is viewed as a temporal average, rather than as a limit. Attention is then focused on partition functions, the theorem of equipartition of energy and an elementary theory of specific heats. In the second part, the Planck derivation of the law of black-body radiation is analysed, presenting in chronological order the Kirchhoff theorem, the Stefan law, the Wien displacement law, the Rayleigh–Jeans formula and the Planck hypothesis. Further topics discussed are the Einstein and Debye quantum models for specific heats of solids. These topics prepare the ground for the introduction of quantum statistical mechanics. The third part is, in fact, devoted to the analysis of identical particles in quantum mechanics.
The conventional understanding of pressure in physics is strictly tied to physical space, representing the force exerted per unit area by particles colliding with boundaries or interacting within a spatial volume. In this paper, we propose a radical extension of thermodynamic concepts to the domain of quantum mechanics by introducing the framework of the Intrinsic Pressure of Quantum States (IPQS). We postulate that quantum states possess an intrinsic internal pressure, denoted as P_psi, which operates independently of physical space volume and is fundamentally defined by the sensitivity of the system energy E to changes in the accessible state space volume V_Hilbert. By formalizing V_Hilbert as the measure of the accessible volume in the underlying Hilbert space or projective state space, we establish a rigorous mathematical foundation that links quantum state configuration changes to thermodynamic work and stability. We explore the thermodynamic analogies of IPQS, derive the equations of state for closed and open quantum systems, and analyze how modifications to state accessibility influence system stability, phase transitions, and quantum fluctuations. Furthermore, we discuss the profound implications of IPQS for quantum information theory, entanglement dynamics, and many-body quantum systems, paving the way toward a unified thermodynamics of quantum states.
In recent years we have witnessed a concentrated effort to make sense of thermodynamics for small-scale systems. One of the main difficulties is to capture a suitable notion of work that models realistically the purpose of quantum machines, in an analogous way to the role played, for macroscopic machines, by the energy stored in the idealisation of a lifted weight. Despite several attempts to resolve this issue by putting forward specific models, these are far from realistically capturing the transitions that a quantum machine is expected to perform. In this work, we adopt a novel strategy by considering arbitrary kinds of systems that one can attach to a quantum thermal machine and defining work quantifiers . These are functions that measure the value of a transition and generalise the concept of work beyond those models familiar from phenomenological thermodynamics. We do so by imposing simple operational axioms that any reasonable work quantifier must fulfil and by deriving from them stringent mathematical condition with a clear physical interpretation. Our approach allows us to derive much of the structure of the theory of thermodynamics without taking the definition of work as a primitive. We can derive, for any work quantifier , a quantitative second law in the sense of bounding the work that can be performed using some non-equilibrium resource by the work that is needed to create it. We also discuss in detail the role of reversibility and correlations in connection with the second law. Furthermore, we recover the usual identification of work with energy in degrees of freedom with vanishing entropy as a particular case of our formalism. Our mathematical results can be formulated abstractly and are general enough to carry over to other resource theories than quantum thermodynamics.
Quantum mechanics and classical statistical mechanics/thermodynamics are both statistical theories, but have key differences. We argue that one such key difference is nonlocality in the quantum world i.e a wavelength proportional to 1/p and a probability of exp(ipx) associated with p versus locality in the classical case. Quantum nonlocality tends to lead to spatial densities with humps even for a particle with a fixed energy in an infinite potential box. Recently it has been suggested in (1) that one may think of a quantum particle in a box with infinite walls in terms of thermodynamical properties e.g. pressure which is taken to be constant throughout the box. From this assumption, one finds that quantum entropy is ln[ Average(nxnx + nyny+nznz)]. For a single quantum state, this seems to imply that entropy is ln[nxnx+nyny+nznz] ((1a)). In a previous note (2), we suggested that quantum entropy for a wavefunction (say an eigenstate) is Integral dx W*(x)W(x) ln[W*W] + Integral dp a*(p)a(p) ln[a*(p)a(p)] ((1b)). We found this is independent of L for a particle in a box with infinite walls so that displacing the wall adiabatically should mean that dS/dL = 0 which it does as L cancels from ((1)). We argue in this note that quantum pressure should be linked to the density humps for a sin(pave x)sin(pave x) wavefunction even though pave is constant at all x points. Thus it seems that one would not consider a constant pressure for sin(pave x)=wavefunction. Furthermore ((1b)) does not yield the same entropy as ((1a)). It is noted in (1) that ((1a)) is an exact expression for entropy and that the Maxwell-Boltzmann distribution entropy approximates it. In this note, we argue that a time reversal balanced scattering reaction leads to the Fermi-Dirac and Bose-Einstein distributions and these both lead to the Maxwell-Boltzmann distribution under certain conditions. (The Maxwell-Boltzmann distribution may also be obtained from time reversal balanced scattering with looser restrictions.)
What is the meaning of the thermodynamical temperature in quantum mechanics? What is its role in the classical limit? What can we say about the interplay between quantum and thermal fluctuations? Can we impose a constant-temperature constraint within dynamical simulations of quantum systems as we do in simulations of classical systems? These questions will be reviewed here. In particular, it will be shown how a quantum version of the celebrated Nos'e constant-temperature dynamics can be defined within the phase space formulation of quantum mechanics.