The methods of information theory are applied to the problem of describing the transport of internal energy along a linear harmonic chain in a stationary state. It is shown that the average energy flux along the chain is a constant of the motion which can be directly incorporated into the canonical formalism. The classical probability distribution in phase space is constructed and used to evaluate the partition function in the thermodynamic limit. Thermodynamic relationships between the undetermined multipliers and the given information are explicitly determined, as are the singlet- and pair-distribution functions in configuration and momentum space and the correlation of the momentum of a specific particle in time. For a given system energy there is an upper bound for the mean rate of energy transport. As this limit is closely approached, both the range and duration of correlations become large. This behavior, as well as some other features of the model, is reminiscent of that near a critical point.
The methods of information theory are applied to the problem of describing the transport of internal energy along a linear harmonic chain in a stationary state. It is shown that the average energy flux along the chain is a constant of the motion which can be directly incorporated into the canonical formalism. The classical probability distribution in phase space is constructed and used to evaluate the partition function in the thermodynamic limit. Thermodynamic relationships between the undetermined multipliers and the given information are explicitly determined, as are the singlet- and pair-distribution functions in configuration and momentum space and the correlation of the momentum of a specific particle in time. For a given system energy there is an upper bound for the mean rate of energy transport. As this limit is closely approached, both the range and duration of correlations become large. This behavior, as well as some other features of the model, is reminiscent of that near a critical point.
When a quantum system is coupled to several heat baths at different temperatures, it eventually reaches a non-equilibrium steady state (NESS) featuring stationary internal heat currents.From one side, these currents are responsible to cause decorehence and produce entropy in the system.However, their existence also induce correlations between different parts of the system.In this work, we explore this two-folded aspect of NESSs.Using phase-space techniques we calculate the Wigner entropy production on general linear networks of harmonic nodes.Working in the ubiquitous limit of weak internal coupling and weak dissipation, we obtain simple closedform expressions for the entropic contribution of each individual quasi-probability current.Our analysis also shows that, it is exclusively the (reversible) internal dynamics which maintain the stationary (irreversible) entropy production.From the informational point of view, we address how to quantify the amount of information that disconnected parts of a quantum chain share in a non-equilibrium steady-state.As we show, this is more precisely captured by the conditional mutual information (CMI), a more general quantifier of tripartite correlations than the usual mutual information.As an application, we apply our framework to the paradigmatic problem of energy transfer through a chain of oscillators subject to self-consistent internal baths that can be used to tune the transport from ballistic to diffusive.We find that the entropy production scales with different power law behaviors in the ballistic and diffusive regimes, hence allowing us to quantify what is the "entropic cost of diffusivity".We also compute the CMI for arbitrary sizes and thus find the scaling rules connecting information sharing and diffusivity.Finally, we discuss how this new perspective in the characterization of non-equilibrium systems may be applied to understand the issue of local equilibration in non-equilibrium states.
We study how energy transport in an integrable system is affected by the spectral densities of heat reservoirs. The model investigated here is the quantum harmonic chain with both ends in contact with two heat reservoirs at different temperatures. The master equation for the reduced density matrix is derived on the assumption that the reservoirs are composed of an infinite number of independent harmonic oscillators. We evaluate temperature profile and energy flux in the stationary state for the master equation and discuss how they depend on the types of spectral densities. When we attach the reservoirs of the same type of spectral density, we find that the temperature profile is independent of the types. On the other hand, when the two reservoirs have different types of spectral densities, the energy profile near the ends of the chain depends on the types. When the coupling is finite, the temperature profile near the ends shows a wide variation of behavior dependent on spectral densities and temperatures of reservoirs. This dependence is discussed with the Fokker-Planck equations obtained in the classical limit.
linearSchrod- of coupled harmonic oscillators is investigated by means of the inger coordinates. Assuming such an initial canonical ensemble that half the system is at temperature T and other half at temperature zero, the correlation functions of particles in the system are calculated as functions of time. Thus, it is shown that the average kinetic and potential energies of each particle in the infinitely large system approach different stationary values in both sides of the impurity atom, and that the system approach a stationary state with a gap of energy distribution along the system at the impurity site. This result agrees with that obtained by Kashiwamura and Teramoto by means of trigonometric eigenfunctions of the dynamical system.The elements of position-velocity correlation matrix are also calculated. The energy flow in the large system is derived from these, and it is shown that even after a sufficiently long time the finite energy flow still exists at every point of the system though there is no temperature gradient at that point.It should be mentioned, however, that the method we used here is easier to calculate and simpler by far than that by means of the trigonometric eigenfunctions of the dynamical system.
The manner in which probability amplitudes of paths sum up to form wave functions of a harmonic oscillator, as well as other, simple 1-dimensional problems, is described. Using known, closed-form, path-based propagators for each problem, an integral expression is written that describes the wave function. This expression conventionally takes the form of an integral over initial locations of a particle, but it is re-expressed here in terms of a characteristic momentum associated with motion between the endpoints of a path. In this manner, the resulting expression can be analyzed using a generalization of stationary-phase analysis, leading to distributions of paths that exactly describe each eigenstate. These distributions are valid for all travel times, but when evaluated for long times they turn out to be real, non-negative functions of the characteristic momentum. For the harmonic oscillator in particular, a somewhat broad distribution is found, peaked at value of momentum that corresponds to a classical energy which in turn equals the energy eigenvalue for the state being described.
An earlier study of quantum-mechanical time has suggested that, from the quantum-theoretic point of view, time is a random variable. The present work explores how such a description of time can be used to bring greater detail into the analysis of the one-dimensional harmonic oscillator. Specifically, the classical equations for harmonic motion are used as a point transformation between phase space and time-energy space, permitting probability distributions in the two spaces to be different representations of the same thing. The familiar energy-quantization hypothesis is supplemented with the hypothesis that quantized energy values are means (averages) of energy distributions characteristic of each energy state. With the probability distribution of time determined from the previously cited study, the desired quantized energy distributions follow from the requirements: (1) that they should be normalized, non-negative, and with means equal to the quantized energy values; (2) that the equivalent representation of distribution in position-momentum space should have marginal densities that are proportional (except near the origin) to the densities in position and momentum, ${|\ensuremath{\phi}(q)|}^{2}$ and ${|\ensuremath{\psi}(p)|}^{2}$, which are postulated in quantum theory. Requirement (2) is a weakening of the Born postulate. In the time-energy space, the probability distributions exclude energies smaller than the mode (energy value with maximum probability). In position-momentum space this exclusion requirement becomes the dynamical restriction: $\frac{{q}^{2}m{\ensuremath{\omega}}^{2}}{2}+\frac{{p}^{2}}{2m}\ensuremath{\ge}\frac{{\ensuremath{\Lambda}}_{n}\ensuremath{\hbar}\ensuremath{\omega}}{2}$, where, for the first three energy states, ${\ensuremath{\Lambda}}_{0}=0$, ${\ensuremath{\Lambda}}_{1}=\frac{3}{2}$, ${\ensuremath{\Lambda}}_{2}=2+\ensuremath{\surd}\frac{3}{2}$. In general terms it is thus shown that the quantized harmonic oscillator can be described as a statistical ensemble of classical oscillators. Some comparisons with work on phase-space formulations of quantum mechanics (especially with the work of Wigner) are made and some aspects of the theory that should submit to experimental verification are pointed out.
We consider two quantum Ising chains initially prepared at thermal equilibrium but with different temperatures and coupled at a given time through one of their end points. In the long-time limit the system reaches a nonequilibrium steady state. We discuss properties of this nonequilibrium steady state and characterize the convergence to the steady regime. We compute the mean energy flux through the chain and show that the heat transport is ballistic. We derive also the large-deviation function for the quantum and thermal fluctuations of this energy transfer.