Guunter Mahler

System and Environment

Since the considerations in chap. 8 suggest that typicality of states or increasing von Neumann entropy may only be found in composite systems, we here formulate a system–environment scenario in some detail. Technically an adequate notation is introduced. We comment on the fact that any thermodynamic scenario comprises some sort of environment. This statement holds if the notion of “environment” is enlarged to include not only systems with which extensive quantities (heat, particles, etc.) may be exchanged but any interacting quantum system. If the system is, e.g., a gas, the environment is or includes at least the vessel that contains the gas. We explain in which sense weak system–environment coupling plays a crucial role in our scenario. We, furthermore, discuss the implications of this weak coupling on a pertinent accessible region. Eventually couplings are classified according to whether or not they allow for energy exchange between system and environment.

Observability of Intensive Variables

The status of the intensive variables like temperature or pressure needs further exploration; two interrelated questions are addressed in this chapter: First we investigate under what conditions temperature can be defined locally. We then discuss basic procedures by which pressure and temperature, though not observables by themselves, could be inferred from Experiment.

Observability of Extensive Variables

The observability of extensive thermodynamic variables is most conveniently related to thermodynamic processes: One can study the change of internal energy subject to different constraints. This leads, e.g., to the concept of work and heat, the later being related to a change of entropy. Fluctuations of work become important in the limit of small systems, i.e., in the domain of “nano-thermodynamics,” and in connection with non-equilibrium.

Typicality of Observables and States

The implementation of the typicality approach to quantum systems as introduced in Chaps.6 and 7 is further elaborated on. Thus, a concrete class of accessible regions as formally introduced in sect. 6 is given. To start with, expectation values of observables are investigated for typicality. We essentially find that typicality can be expected for observables which are defined on high-dimensional Hilbert spaces, but feature bound spectra within the accessible state space. The typical values of the observable are in accord with Boltzmann’s principle of “equal a priori probabilities.” Furthermore, typicality of a state rather than for an observable is introduced. We find that, while there may very well be such typicality of some observables there is no typicality of states for non-composite systems. This points toward the investigation of composite systems.

Open System Approach to Transport1

In this section yet another approach to transport is examined. Rather than analyzing spatial energy density dynamics as done in the previous section, we now aim at investigating a stationary non-equilibrium state. This state is induced through the local coupling of reservoirs with different temperatures at either end of the system. The system is a chain of two-level systems coupled according to the Heisenberg model. The baths are modeled according to standard open system theory (see [3–6]). The resulting equations are numerically solved using a stochastic unraveling scheme.

Brief Review of Relaxation and Transport Theories

In this chapter we will rather briefly mention well-established approaches to relaxation and transport behavior in quantum systems. This is not meant to be a comprehensive review of all approaches concerning those complex fields of theoretical physics, but an overview of concepts and central ideas. To learn more about the discussed topics, the interested reader is referred to the substantial secondary literature which is cited in this chapter.

Projective Approach to Dynamical Transport

In this chapter we analyze transport properties of spatially structured, simplified quantum systems using the time-convolutionless (TCL) method introduced in Chap. 18. Therefore the system is initially prepared in a non-equilibrium state, ie, a non-equilibrium distribution of energy or heat, for example. After the relaxation process, the system will be at a global equilibrium in the sense of a uniform energy distribution but not in the sense of a maximum global von Neumann entropy. From the concrete dynamics of this decay we extract information about the transport properties of the model.

Dynamics and Averages in Hilbert Space

In the previous chapter the concept of typicality has been introduced for a rather abstract space of the states of a system. Also the properties of the dynamics which are imperative for the applicability of the concept have been denoted quite formally. Here we introduce the Hilbert space of pure quantum states as our concrete state space and formulate a representation of Hilbert space. Within this representation the dynamics as described by the Schrodinger equation indeed meets the above requirements. Furthermore the averages and variances that occur in the context of typicality are mathematically concretized for this quantum case and thus accordingly called Hilbert space average and Hilbert space variance, respectively.

Finite Systems as Thermostats

Having derived the formal mathematical background of projection operator techniques (TCL) and the Hilbert space average method (HAM) in the last chapter, we will now use these techniques to analyze a system–reservoir scenario. The above introduced partitioning scheme into a relevant part and the rest will be used to consider the relaxation of a small quantum system, e.g., a two-level atom to equilibrium. Contrary to the standard bath scenario the atom is coupled to a finite environment here, i.e., another not too big quantum system. We show, by comparing with the (numerically) exact dynamics given by the Schrodinger equation, that the projective approach may work or fail, depending on the projector chosen.

Outline of the Present Approach

As already indicated in previous chapters we intend to explain the validity of the laws of thermodynamics on the basis of quantum dynamics as given by the Schrodinger equation. In this chapter we provide the background of this approach which does not rely on ergodicity, mixing, etc., but on the concept of “typicality.” The importance of a system–environment scheme for a concise definition of entropy and the role of entanglement in this context is discussed. Furthermore we investigate the nature of thermodynamical uncertainties (often referred to as “fluctuations”). Within this quantum version of the typicality approach these uncertainties appear as quantum uncertainties and are thus of fundamental nature.