Statistical Physics I

It is shown that a unique measure of volume is associated with any statistical ensemble, which directly quantifies the inherent spread or localisation of the ensemble. It is applicable whether the ensemble is classical or quantum, continuous or discrete, and may be derived from a small number of theory-independent geometric postulates. Remarkably, this unique ensemble volume is proportional to the exponential of the ensemble entropy, and hence provides a novel geometric characterisation of the latter quantity. Applications include unified volumebased derivations of the Holevo and Shannon bounds in quantum and classical information theory, a precise geometric interpretation of thermodynamic entropy for equilibrium ensembles, a geometric derivation of semi-classical uncertainty relations, a new means for defining classical and quantum localization for arbitrary evolution processes, a geometric interpretation of relative entropy, and a new proposed definition for the spot-size of an optical beam. Advantages of ensemble volume over other measures of localization (root-mean-square deviation, Renyi entropies, and inverse participation ratio) are discussed.PACS Numbers: 03.65. Bz, 05.45+b, 42.60.Jf I INTRODUCTIONThis paper has two main goals. The first is to demonstrate that for any ensemble, whether classical, quantum, discrete or continuous, there is essentially only one measure of the "volume" occupied by the ensemble which is compatible with basic geometric notions. This ensemble volume is thus a preferred and universal choice for characterising what is variously referred to as the spread, dispersion, uncertainty, or localisation of an ensemble. Remarkably, the derived "ensemble volume" turns out to be proportional to the exponential of the entropy of the ensemble. A by-product of the first goal is thus a new universal characterisation of ensemble entropy, based on geometric notions. Indeed, a number of properties of ensemble entropy turn out to have simple geometric interpretations. The universal nature of the characterisation is of particular interest: the only previous contextindependent interpretation of ensemble entropy to date (and hence applicable in particular to ensembles described by continuous probability distributions) appears to be as a somewhat vague measure of uncertainty or randomness. 2The second goal is to apply "ensemble volume" to a wide range of contexts in which ensembles appear. The applications demonstrate not only the advantages of ensemble volume over other measures of spread, but also to some extent why it is that ensemble entropy makes a natural appearance in contexts as diverse as statistical mechanics, information theory, chaos, and quantum uncertainty relations. Some results have been briefly reported elsewhere [1]. Here important details and extensions are given, as well as a number of new results.The work reported here was originally motivated by several connections between volume and information. Shannon proved an upper bound on information transfer, via classical signals s...

show abstract

“….}. However, continuous classical ensembles violate the third law [5] and K(Γ) remains arbitrary in this case (but see Sec. IV.B below).…”

Section: A Statistical Mechanicsmentioning

confidence: 99%

Universal geometric approach to uncertainty, entropy, and information

Hall

1999

Phys. Rev. A

145

View full text Add to dashboard Cite

show abstract

“…As a first step, we specify the functional dependence of the total energy E of the system, which needs to be an extensive variable [50,51]:…”

Section: Thermodynamic Relationsmentioning

confidence: 99%

Hydrodynamic description of elastic or viscoelastic composite materials: Relative strains as macroscopic variables

Menzel

2016

Phys. Rev. E

View full text Add to dashboard Cite

One possibility to adjust material properties to a specific need is to embed units of one substance into a matrix of another substance. Even materials that are readily tunable during operation can be generated in this way. In (visco)elastic substances, both the matrix material as well as the inclusions and/or their immediate environment can be dynamically deformed. If the typical dynamic response time of the inclusions and their surroundings approach the macroscopic response time, their deformation processes need to be included into a dynamic macroscopic characterization. Along these lines, we present a hydrodynamic description of (visco)elastic composite materials. For this purpose, additional strain variables reflect the state of the inclusions and their immediate environment. These additional strain variables in general are not set by a coarse-grained macroscopic displacement field. Apart from that, during our derivation, we also include the macroscopic variables of relative translations and relative rotations that were previously introduced in different contexts. As a central point, our approach reveals and classifies the importance of a new macroscopic variable: relative strains. We analyze two simplified minimal example geometries as an illustration.

show abstract

“…Note that all thermodynamic quantities involved, such as entropy, internal energy (5), temperature (6), heat (9) and work (10), are defined in the quasi-static limit in such a way that Eq. (8) turns into the Gibbsian fundamental form:…”

Section: Carnot Processmentioning

confidence: 99%

“…For example, following the formal definition [9,10], we introduce the temperature T as the conjugate variable to the entropy S:…”

Section: Thermodynamic Quantitiesmentioning

confidence: 99%