A Method of Calculation of Electrical Conductivity

Nakano, Huzio

doi:10.1143/ptp.15.77

Cited by 109 publications

(36 citation statements)

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“…In the previous paper [1], the author studied the irreversibility and entropy production in transport phenomena [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], using the idea of the symmetry separation of the density matrix in the von Neumann equation.…”

Section: Kubo-type Formulation Of the Non-equilibrium Density Matrixmentioning

confidence: 99%

“…In the first paper of this series [1], the present author has derived the irreversible entropy production in transport phenomena [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] starting from the von Neumann equation, as Kubo et al [2][3][4] formulated the linear response scheme. A new aspect of the present author's theory [1] is that the entropy operator S is defined using the equilibrium density matrix ρ eq as S = −k B log ρ eq = (H 0 − F 0 )/T (1) with F 0 = −k B T log Tr exp(−βH 0 ) and that the entropy production is obtained thereby from the symmetric part ρ s (t) of the density matrix [1]:…”

Section: Introductionmentioning

confidence: 99%

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Irreversibility and entropy production in transport phenomena, II: Statistical–mechanical theory on steady states including thermal disturbance and energy supply

Suzuki

2012

Physica A: Statistical Mechanics and its Applications

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Some general aspects of nonlinear transport phenomena are discussed on the basis of two kinds of formulations obtained by extending Kubo's perturbational scheme of the density matrix and Zubarev's non-equilibrium statistical operator formulation. Both formulations are extended up to infinite order of an external force in compact forms and their relationship is clarified through a direct transformation. In order to make it possible to apply these formulations straight-forwardly to thermal disturbance, a mechanical formulation of it is given (in a more convenient form than Luttinger's formulation) by introducing the concept of a thermal field E T which corresponds to the temperature gradient and by defining its conjugate heat operator A H = j h j r j for a local internal energy h j of the thermal particle j. This yields a transparent derivation of the thermal conductivity κ of the Kubo form and the entropy production (dS/dt) irr = κE 2 T /T . Mathematical aspects on the nonequilibrium density-matrix will also be discussed. In Paper I (Physica A 390(2011)1904), the symmetry-separated von Neumann equation with relaxation terms extracting generated heat outside the system was introduced to describe the steady state of the system. In this formulation of the steady state, the internal energy H 0 t is time-independent but the field energy H 1 t (= − A t ·F ) decreases as time t increases. To overcome this problem, such a statistical mechanical formulation is proposed here as includes energy supply to the system from outside by extending the symmetry-separated von Neumann equation given in Paper I. This yields a general theory based on the density-matrix formulation on a steady state with energy supply inside and heat extraction outside and consequently with both H 0 t and H 1 t constant. Furthermore, this steady state gives a positive entropy production. The present general formulation of the current yields a compact expression of the time derivative of entropy production, which yields the plausible justification of the principle of minimum entropy production in the steady state even for nonlinear responses.

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Section: Kubo-type Formulation Of the Non-equilibrium Density Matrixmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Irreversibility and entropy production in transport phenomena, II: Statistical–mechanical theory on steady states including thermal disturbance and energy supply

Suzuki

2012

Physica A: Statistical Mechanics and its Applications

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“…The basic problems of irreversibility and transport phenomena have been discussed for many years from many viewpoints [1][2][3][4][5][6][7][8][9][10][11][12]. Boltzmann's H-theorem shows the first attempt to explain the broken symmetry of entropy change in time, namely irreversibility, even though it is based on stochastic equations (such as the Boltzmann equation).…”

Section: Introductionmentioning

confidence: 99%

“…Kubo and Tomita [8] succeeded, for the first time, in formulating magnetic linear responses on the basis of the von Neumann equation. Nakano [9] applied Kubo-Tomita's method to the problem of electric conduction, though the two problems had been regarded to be different in the sense that the former is reduced to an equilibrium problem for zero frequency limit, but the latter is still in non-equilibrium even for the static limit [10]. Kubo [4] established a general theory of linear responses including the magnetic response and electric conduction, as is briefly reviewed in Section 2.…”

Section: Introductionmentioning

confidence: 99%

Irreversibility and entropy production in transport phenomena I

Suzuki

2011

Physica A: Statistical Mechanics and its Applications

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The linear response framework was established a half-century ago, but no persuasive direct derivation of entropy production has been given in this scheme. This long-term puzzle has now been solved in the present paper. The irreversible part of the entropy production in the present theory is given by (dS/dt) irr = (dU/dt)/T with the internal energy U(t) of the relevant system. Here, U(t) = H 0 t = TrH 0 ρ(t) for the Hamiltonian H 0 in the absence of an external force and for the density matrix ρ(t). As is well known, we have (dS/dt) irr = 0 if we use the linear-order density matrix ρ lr (t) = ρ 0 + ρ 1 (t). Surprisingly, the correct entropy production is given by the second-order symmetric term ρ 2 (t) as (dS/dt) irr = (1/T )TrH 0 ρ ′ 2 (t). This is shown to agree with the ordinary expression J · E/T = σE 2 /T in the case of electric conduction for a static electric field E, using the relations TrH 0 ρ ′ 2 (t) = −TrḢ 1 (t)ρ 1 (t) = TrȦ · Eρ 1 (t) = J · E (Joule heat), which are derived from the second-order von Neumann equation ihdρ 2 (t)/dt = [H 0 , ρ 2 (t)] + [H 1 (t), ρ 1 (t)]. Here H 1 (t) denotes the partial Hamiltonian due to the external force such as H 1 (t) = −e j r i · E ≡ −A · E in electric conduction. Thus, the linear response scheme is not closed within the first order of an external force, in order to manifest the irreversibility of transport phenomena. New schemes of steady states are also presented by introducing relaxation-type (symmetry-separated) von Neumann equations. The concept of stationary temperature T st is introduced, which is a function of the relaxation time τ r characterizing the rate of extracting heat outside from the system. The entropy production in this steady state depends on the relaxation time. A dynamical-derivative representation method to reveal the irreversibility of steady states is also proposed. The present derivation of entropy production is directly based on the first principles of using the projected density matrix ρ 2 (t) or more generally symmetric density matrix ρ sym (t), while the previous standard argument is due to the thermodynamic energy balance. This new derivation clarifies conceptually the physics of irreversibility in transport phenomena, using the symmetry of non-equilibrium states, and this manifests the duality of current and entropy production.

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“…Furthermore, we construct the optimized/renormalized perturbation theory (See e.g. [44,45] and references therein for references of optimized perturbation theory) for real-time dynamics and derive the dissipative transport together with the Green-Kubo formula for the transport coefficients [46][47][48].…”

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confidence: 99%

Nonrelativistic Hydrodynamics from Quantum Field Theory: (I) Normal Fluid Composed of Spinless Schrödinger Fields

Hongo

2019

J Stat Phys

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We provide a complete derivation of hydrodynamic equations for nonrelativistic systems based on quantum field theories of spinless Schrödeinger fields, assuming that an initial density operator takes a special form of the local Gibbs distribution. The constructed optimized/renormalized perturbation theory for real-time evolution enables us to separately evaluate dissipative and nondissipative parts of constitutive relations. It is shown that the path-integral formula for local thermal equilibrium together with the symmetry properties of the resulting action-the nonrelativistic diffeomorphism and gauge symmetry in the thermally emergent Newton-Cartan geometry-provides a systematic way to derive the nondissipative part of constitutive relations. We further show that dissipative parts are accompanied with the entropy production operator together with two kinds of fluctuation theorems by the use of which we derive the dissipative part of constitutive relations and the second law of thermodynamics. After obtaining the exact expression for constitutive relations, we perform the derivative expansion and derive the first-order hydrodynamic (Navier-Stokes) equation with the Green-Kubo formula for transport coefficients.

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A Method of Calculation of Electrical Conductivity

Cited by 109 publications

References 0 publications

Irreversibility and entropy production in transport phenomena, II: Statistical–mechanical theory on steady states including thermal disturbance and energy supply

Irreversibility and entropy production in transport phenomena, II: Statistical–mechanical theory on steady states including thermal disturbance and energy supply

Irreversibility and entropy production in transport phenomena I

Nonrelativistic Hydrodynamics from Quantum Field Theory: (I) Normal Fluid Composed of Spinless Schrödinger Fields

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