Jochen Gemmer

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.

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.

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.

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.