We define the entropy operator as the negative of the logarithm of the density matrix, give a prescription for extracting its thermodynamically measurable part, and discuss its dynamics. For an isolated system we derive the first, second, and third laws of thermodynamics. For weakly coupled subsystems of an isolated system, an expression for the long-time limit of the expectation value of the rate of change of the thermodynamically measurable part of the entropy operator is derived and interpreted in terms of entropy production and entropy transport terms. The interpretation is justified by comparison to the known expression for the entropy production in an aged classical Markovian system with Gaussian fluctuations and by a calculation of the current-induced entropy production in a conductor with electron-phonon scattering.
We define the entropy operator as the negative of the logarithm of the density matrix, give a prescription for extracting its thermodynamically measurable part, and discuss its dynamics. For an isolated system we derive the first, second, and third laws of thermodynamics. For weakly coupled subsystems of an isolated system, an expression for the long-time limit of the expectation value of the rate of change of the thermodynamically measurable part of the entropy operator is derived and interpreted in terms of entropy production and entropy transport terms. The interpretation is justified by comparison to the known expression for the entropy production in an aged classical Markovian system with Gaussian fluctuations and by a calculation of the current-induced entropy production in a conductor with electron-phonon scattering.
In this thesis we develop an approach to nonequilibrium quantum-statistical mechanics which is based on the consideration of the thermodynamic entropy as being a quantum observable with an associated hermitian operator and with its own equation of motion, from which the rate of entropy production can be studied. The relationship of this quantum observable---the expectation value of which is proved to obey the laws of thermodynamics and is thus called the thermodynamic entropy---with heat dissipation in quantum many-body systems is investigated in detail. After showing how the classical theory of nonequilibrium thermodynamics is obtained from our entropy-production formalism in the limit of very weakly coupled subsystems of a larger isolated quantum system, the solution of the equation of motion for the thermodynamic entropy operator is formally obtained and applied to the case of electrons interacting with phonons and being driven by an external electric field, arriving at an explicit expression for the Joule heat without any a priori consideration of the rate of change of the energy of the system. Finally, the formalism is applied to solve a puzzle introduced with the most basic model for atom-light interactions: the Jaynes-Cummings model. Without using the correct thermodynamic arguments implied by our entropy-production theory, this model leads to the conclusion that a stream of two-level atoms sent to a cavity filled with monochromatic photons---in the so called one-atom maser configuration---thermalize, under off-resonant conditions, to a temperature different from that of the photons, casting doubts on the validity of the principle of conservation of energy.
Bridging the second law of thermodynamics and microscopic reversible dynamics has been a longstanding problem in statistical physics. Here, we address this problem on the basis of quantum many-body physics, and discuss how the entropy production saturates in isolated quantum systems under unitary dynamics. First, we rigorously prove that the entropy production does indeed saturate in the long time regime, even when the total system is in a pure state. Second, we discuss the non-negativity of the entropy production at saturation, implying the second law of thermodynamics. This is based on the eigenstate thermalization hypothesis, which states that even a single energy eigenstate is thermal. We also numerically demonstrate that the entropy production saturates at a non-negative value even when the initial state of a heat bath is a single energy eigenstate. Our results reveal fundamental properties of the entropy production in isolated quantum systems at late times.
Thermodynamics is a phenomenological theory describing the energy conversion of work and heat. At its origins thermodynamics was developed in order to understand and improve heat engines. In conventional thermodynamics, however, only such processes are completely describable which are slow enough to keep the system of interest in an equilibrium state with its thermal environment at all times. On the contrary, all real physical processes are accompanied by non-equilibrium phenomena. These are mathematically described with the help of the irreversible entropy production, i.e. the dissipated work divided by the temperature of the bath. The second law of thermodynamics merely states that the irreversible entropy production is on average always positive. The modern trend of miniaturization, however, leads to smaller and smaller devices. On short length scales thermal noise as well as quantum fluctuations become important. Thus, usual thermodynamic quantities as work and heat acquire stochastic nature. Moreover, in the quantum regime a completely new theory had to be invented, since classical notions of work and heat are no longer valid. The present dissertation contributes to this prevailing field by the derivation of analytical expressions for the entropy production in open and closed quantum system far from thermal equilibrium. The theoretical treatment is motivated by an experimental point of view. The obtained results were derived from the reduced dynamics of the quantum mechanical system, which may be coupled to a thermal bath. To this end, it was dealt with methods and approaches of statistical physics, conventional thermodynamics, quantum information theory and the theory of open quantum systems. In this dissertation the most important physical systems are treated. First, an isolated system is considered before later on a heat bath is included. The limit of weak as well as of strong coupling are analyzed in detail. For isolated systems it turns out that a geometric approach is able to capture the thermodynamic properties, completely. To this end, the irreversible entropy production is sharply estimated from below by the angle between the real non-equilibrium state and the corresponding equilibrium one. Further, it is shown that the Heisenberg uncertainty relation for energy and time can be formulated more precisely in terms of that angle. As experimental system cold ion traps are introduced, in which the theoretical predictions can be verified. For weakly coupled systems, then, the irreversible entropy production is identified and a fluctuation theorem is derived by merely thermodynamic arguments. As experimental system we, again, propose the cold ion traps, where the heat bath can be simulated with the help of LASER light. In the limit of strong coupling a semiclassical description of the reduced quantum system is discussed. Again, an analytically exact expression for the entropy production and the corresponding fluctuation theorem is derived. A physical system, in which strong coupling to the environment can be analyzed, is given by Josephson junctions. This dissertation discusses the entropy production for isolated, weakly coupled, and overdamped quantum systems. For these physical regimes analytical relation are derived and experimental systems proposed, with which the predictions can be verified.
We discuss fundamental features of the local expression for the entropy production rate of the nonequilibrium quantum dynamics of open systems and its relations to memory effects and the spectrum of the generator of the dynamics. Defining the entropy production rate as negative rate of change of the relative entropy with respect to an instantaneous fixed point, it is shown that positivity of the entropy production rate for all possible initial states implies that the real parts of the eigenvalues of the time-local generator for the quantum master equation are always negative. It is further demonstrated that Markovian dynamics, identified as P-divisibility of the quantum dynamical map, implies positivity of entropy production rate, thus providing a kind of generalized second law in the nonequilibrium regime. We also prove by means of the counterexample of a phase covariant quantum master equation that the converse of this statement is not true, i.e., there are non-Markovian dynamics for which the entropy production rate is always positive. Thus, we conclude that the emergence of negative entropy production rates is a sufficient but not necessary condition for non-Markovianity of the quantum dynamics. Finally, we also consider a recently introduced map-based notion of entropy production and show the equivalence between its positivity and Markovianity for general finite-dimensional systems.
We consider the temporal evolution of the entropy S of an isolated system from an initial non-equilibrium state 2· of entropy S = S i to the equilibrium state of maximum entropy S = S max > S i. The application of usual density matrix theory of quantum mechanics leads us to the surprising result P = dS/dt = 0. The entropy does not change in time. This result is valid in case of the equilibrium state. However it is wrong for a non-equilibrium state, because the entropy of the isolated system could not spontaneously increase from S i to S max by entropy production P > 0. This contradicts the second law of thermodynamics. We avoid this discrepancy by changing from quantum mechanics to quantum thermodynamics. Now the entropy production P = dS/dt and the time t are the relevant observables. In order to treat the temporal development of S and P, we at first prepare an initial state i of entropy production P i at time t < 0. Subsequently at ϑ t ≥ 0 the initial state i interacts with a continuum of final states f, which transfers the entropy production from i to f. The temporal evolution of the entropy production P(t) of our system is treated by means of the time-dependent perturbation theory. For that reason we split the operator of entropy production P = P 0 + W into two parts. The operator P 0 describes the unperturbed basic vectors of entropy production | i > and | f >, while the perturbation operator W determines the interaction between | i > and | f >. We expand the time-dependent state function < t | ψ > of the isolated system into the complete set of stationary eigenvectors | i > and | f > and calculate the time-dependent expansion coefficients c i(t) and c(f, t) by means of the dynamic equation of quantum thermodynamics − i k (d/dt) < t | ψ > = < t | P | ψ >. Here k means the Boltzmann constant, which is characteristic of quantum thermodynamics and represents the atomic entropy unit. The expansion coefficients are given by a system of coupled differential equations, from which we deduce an integral-differential equation for the coefficient c i(t) of the initial state i. Unfortunately it has been impossible to obtain an analytic solution of the complicated equation. However approximations are available. Accordingly, in the vicinity of the equilibrium state, the probability of still finding the initial state i of entropy production P i at time t is given by ω i(t) = | c i(t) |2 = exp (−Lt). The rate constant L ≥ σ depends on the density ρ(P f) of final states f of entropy production P f near by P i, and on the squared modulus of the perturbation matrix elements W if = < i | W | f >, which determines the interaction between | i > and | f >. The entropy production P(t) = P i ω i(t) = P i exp (−ϑLt) of the isolated system decreases exponentially and irreversibly with proceeding time t. Its rate-determining step is the transfer of entropy production from i to f. The integral over the entropy production of the final states f vanishes. Therefore the continuum states f do not contribute to the total entropy production P(t). The temporal evolution of the entropy S(t) can easily be calculated by integration of P(t). In conclusion, the entropy production of a non-equilibrium state is finite P(t) > 0 and only vanishes in the case of the equilibrium state P(∞) = 0. Thus, quantum thermodynamics agrees with the second law of thermodynamics.
We consider the temporal evolution of an isolated system of particles from a non-equilibrium state of entropy S = S′ to the equilibrium state of maximum entropy S = S max, S′ ≤ S max. The application of usual density matrix theory to the temporal development of S leads us to the unexpected result P = dS/dt = 0. The entropy does not change in time. This result is valid in case of the equilibrium state. However it is wrong for a non-equilibrium state, which could not irreversibly change from S′ to S max by entropy production P > 0. This contradicts the second law of thermodynamics. The paradoxial result can be traced back to the Liouville theorem. To overcome this difficulty, we introduce the t–P space. The time t and the entropy production P are observables in quantum thermodynamics. They are defined by the eigenvalue equations t|t〉 = t|t〉 and P|P〉 = P|P〉. The operators t and P do not commute. We have a non-vanishing commutator, [t,P] = ik, and consequently a t–P uncertainty relation, ΔtΔP ≥ k/2. The Boltzmann constant k is characteristic of quantum thermodynamics and represents the atomic entropy unit. — The time evolution operator U(τ) = exp[−(i/k)τP] shifts the time t by τ, U(τ)|t〉 = |t + τ〉. The generator of τ is the operator P of entropy production. Then the eigenvector |P〉 is also an eigenket of U(t), and it follows with progressing time τ ≥ 0 that τP ≥ 0 and hence P = dS/dt ≥ 0. This is the second law of thermodynamics, which is correctly reproduced in quantum thermodynamics. — Exchanging −τ for +τ replaces the equations of forward time +τ by the equations of backward time −τ. Independent of the direction of the arrow of time, the entropy of the system always increases or remains constant in the limit of the equilibrium state, dS ≥ 0. This is characteristic of irreversibility, for the system does not return to its initial state on time reversal. — The t–P uncertainty relation allows to discuss the temporal evolution of the expectation value 〈P〉 of entropy production and the lifetime Δt of the transient states, the system passes through on approaching the equilibrium state. In case of a non-equilibrium state 〈P〉, ΔP, and Δt are finite. On approaching the equilibrium state, 〈P〉 and ΔP decrease, consequently Δt increases. Finally, in the case of equilibrium (eq), we have 〈P〉eq = 0, ΔP eq = 0, and Δt eq → ∞. The lifetime of the equilibrium state is unlimited, because the entropy production 〈P〉 and the uncertainty of entropy production ΔP exactly vanish. The temporal evolution of the system can be described by means of a phenomenological equation.