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We present a general nonequilibrium Green's function formalism for modeling heat transfer in systems characterized by linear response that establishes the formal algebraic relationships between phonon and radiative conduction, and reveals how upper bounds for the former can also be applied to the latter. We also propose an extension of this formalism to treat systems susceptible to the interplay of conductive and radiative heat transfer, which becomes relevant in atomic systems and at nanometric and smaller separations where theoretical descriptions which treat each phenomenon separately may be insufficient. We illustrate the need for such coupled descriptions by providing predictions for a low-dimensional system of carbyne wires in which the total heat transfer can differ from the sum of its radiative and conductive contributions. Our framework has ramifications for understanding heat transfer between large bodies that may approach direct contact with each other or that may be coupled by atomic, molecular, or interfacial film junctions., Comment: 16 pages, 2 figures, 1 table, 2 appendices

Near-field radiative heat transfer between bodies at the nanoscale can surpass blackbody limits on thermal radiation by orders of magnitude due to contributions from evanescent electromagnetic fields, which carry no energy to the far-field. Thus far, principles guiding explorations of larger heat transfer beyond planar structures have assumed utility in surface nanostructuring, which can enhance the density of states, and further assumed that such design paradigms can approach Landauer limits, in analogy to conduction. We derive fundamental shape-independent limits to radiative heat transfer, applicable in near- through far-field regimes, that incorporate material and geometric constraints such as intrinsic dissipation and finite object sizes, and show that these preclude reaching the Landauer limits in all but a few restrictive scenarios. Additionally, we show that the interplay of material response and electromagnetic scattering among proximate bodies means that bodies which maximize radiative heat transfer actually maximize scattering rather than absorption. Finally, we compare our new bounds to existing Landauer limits, as well as limits involving bodies maximizing far-field absorption, and show that these lead to overly optimistic predictions. Our results have ramifications for the ultimate performance of thermophotovoltaics and nanoscale cooling, as well as related incandescent and luminescent devices., 12 pages including appendices, 1 figure; SM and PSV contributed equally

Recent experimental advances probing coherent phonon and electron transport in nanoscale devices at contact have motivated theoretical channel-based analyses of conduction based on the nonequilibrium Green's function formalism. The transmission through each channel has been known to be bounded above by unity, yet actual transmissions in typical systems often fall far below these limits. Building upon recently derived radiative heat transfer limits and a unified formalism characterizing heat transport for arbitrary bosonic systems in the linear regime, we propose new bounds on conductive heat transfer. In particular, we demonstrate that our limits are typically far tighter than the Landauer limits per channel and are close to actual transmission eigenvalues by examining a model of phonon conduction in a 1-dimensional chain. Our limits have ramifications for designing molecular junctions to optimize conduction., Comment: 10 pages, 2 figures, 2 appendices

We construct a kinetic equation modeling the behavior of degenerate quantum Bose gases whose collision rate depends on the momentum of elementary excitations. We consider the case where the phonon component is the decisive factor in the elementary excitations. We analytically solve the half-space boundary value problem of the temperature jump at the boundary of the degenerate Bose gas in the presence of a Bose -- Einstein condensate., Comment: 9 pages, 2 figures

A kinetic solver is developed for the initial and boundary value problems (IBVP) of the symmetric hyperbolic moment system. This nonlinear system of equations is related to the heat conduction in solids at low temperatures. The system consists of a conservation equation for the energy density e and a balance equation for the heat flux 혘푖, where 푒 and 혘푖 are the four basic fields of the theory. We use kinetic flux vector splitting (KFVS) scheme to solve these equations in one and two space dimensions. The flux vectors of the equations are splitted on the basis of the local equilibrium distribution of phonons. The resulting computational procedure is efficient and straightforward to implement. The second order accuracy of the scheme is acheived by using MUSCL-type reconstruction and min-mod nonlinear limitters. The solutions exhibit second order accuracy, and satisfactory resolution of gradients with no spurious oscillations. The secheme is extended to the two-dimensional case in a usual dimensionally split manner. In order to prescribe the boundary data we need the knowledge of the 푒 and 혘푖. However, in experiments only one of the quantities can be controlled at the boundary. This problem is removed by using a continuity condition. It turned out that after some short time energy and heat flux are related to each other according to Rankine Hugoniot jump relations. To illustrate the performance of the KFVS scheme, we perform several one- and two-dimensional test computations. For the comparison of our results we use high order central schemes. The present study demonstrates that this kinetic method is effective in handling such problems.

Pure quantum mechanics can be formulated as a Hamiltonian system in terms of the Liouville equation for the density matrix. Dissipative effects are modeled via coupling to a macroscopic system, where the coupling operators act via commutators. Following Ottinger (2010) we use the GENERIC framework to construct thermodynamically consistent evolution equations as a sum of a Hamiltonian and a gradient-flow contribution, which satisfy a particular non-interaction condition: q = J(q)DE(q) + K(q)DS(q). We give three applications of the theory. First, we consider a finite-dimensional quantum system that is coupled to a finite number of simple heat baths, each of which is described by a scalar temperature variable. Second, we model quantum system given by a one-dimensional Schrodinger operator connected to a onedimensional heat equation on the left and on the right. Finally, we consider thermoopto-electronics, where the Maxwell-Bloch system of optics is coupled to the energydrift-diffusion system for semiconductor electronics.

Pure quantum mechanics can be formulated as a Hamiltonian system in terms of the Liouville equation for the density matrix. Dissipative effects are modeled via coupling to a macroscopic system, where the coupling operators act via commutators. Following Ottinger (Phys. Rev. A 82:052119(11), 2010) we use the GENERIC framework (General Equations for Non-Equilibrium Reversible Irreversible Coupling) to construct thermodynamically consistent evolution equations as a sum of a Hamiltonian and a gradient-flow contribution, which satisfy a particular non-interaction condition: $$\displaystyle{\dot{q} = \mathbb{J}(q)\mathrm{D}\mathcal{E}(q) + \mathbb{K}(q)\mathrm{D}\mathcal{S}(q).}$$ One of our models couples a quantum system to a finite number of heat baths each of which is described by a time-dependent temperature. The dissipation mechanism is modeled via the canonical correlation operator, which is the inverse of the Kubo–Mori metric for density matrices and which is strongly linked to the von Neumann entropy for quantum systems. Thus, one recovers the dissipative double-bracket operators of the Lindblad equations but encounters a correction term for the consistent coupling to the dissipative dynamics. For the finite-dimensional and isothermal case we provide a general existence result and discuss sufficient conditions that guarantee that all solutions converge to the unique thermal equilibrium state. Finally, we compare our gradient flow structure for quantum systems with the Wasserstein gradient flow for the Fokker–Planck equation and the entropy gradient flow for reversible Markov chains.

When highly viscous oil is produced at low temperatures, large pressure drops will significantly decrease production rate. One of possible solutions to this problem is heating of oil well by hot water recycling. We construct and analyze a mathematical model of oil-well heating composed of three linear parabolic PDE coupled with one Volterra integral equation. Further on we construct numerical method for the model and present some simulation results., Comment: 9 pages, 2 figures, Conference on Applied Mathematics and Scientific Computing, Brijuni, Croatia, June 23-27, 2003

In this article we present the first and second order numerical schemes for the solution of initial value problems of the Boltzmann-Peierls equation (BPE). We also modify the numerical schemes for the solution of initial and boundary value problems (IBVP) of its derived hyperbolic moment system. BPE is an integro-differential equation which describes the evolution of heat in crystalline solids at very low temperatures. The BPE describes the evolution of the phase density of a phonon gas. The corresponding entropy density is given by the entropy density of a Bose-gas. We derive a reduced three-dimensional kinetic equation which has a much simpler structure than the original BPE, while it still retain all the properties of the original BPE. Using special coordinates, we get a further reduction of the kinetic equation in one space dimension. We introduce the discrete-velocity model of the reduce BPE in one space dimension. This discrete-velocity model can be discretized in space and time by using finite volume schemes. We derive both first and second order explicit upwind and central schemes for the discrete-velocity kinetic equation as well as for the derived moment system. We use the kinetic approach in order to prescribe boundary conditions for the IBVP of the moment system. Several numerical test cases are considered in order to validate the theory.