2.1.2. Thermodynamical Characterisation

This brings us to the proposal of characterising a generalised Gibbs state based on a constrained maximisation of information (Shannon or von Neumann) entropy [14–16], along the lines advocated by Jaynes [19,20] purely from the perspective of evidential statistical inference. Jaynes' approach is fundamentally different from other more traditional ones of statistical physics, as is the thermodynamical characterisation, compared with the others outlined above, which is exemplified in the following. It is thus a new proposal for background independent equilibrium [14,25], which has the potential of incorporating also the others as special cases, from the point of view of constructing a Gibbs state.

Consider a macroscopic system with a large number of constituent microscopic degrees of freedom. Our (partial) knowledge of its macrostate is given in terms of a finite set of averages { O*a* = *Ua*} of the observables we have access to. Jaynes suggests that a fitting probability estimate (which, once known, will allow us to infer also the other observable properties of the system) is not only one that is compatible with the given observations, but also that which is least-biased in the sense of not assuming any more information about the system than what we actually have at hand (namely, {*Ua*}). In other words, given a limited knowledge of the system (which is always the case in practice for any macroscopic system), the least-biased probability distribution compatible with the given data should be preferred. As shown below, this turns out to be a Gibbs distribution with the general form *e*<sup>−</sup> ∑*a βa*O*a* 

Let Γ be a finite-dimensional phase space (be it extended or reduced), and on it consider a finite set of smooth real-valued functions O*<sup>a</sup>*. Denote by *ρ* a smooth statistical density (real-valued, positive and normalised function) on Γ, to be determined. Then, the prior on the macrostate gives a finite number of constraints, 

$$
\langle \mathcal{O}\_a \rangle\_{\rho} = \int\_{\Gamma} d\lambda \,\,\rho \,\mathcal{O}\_a = \mathcal{U}\_a \tag{1}
$$

where *dλ* is a Liouville measure on Γ, and the integrals are taken to be well-defined. Further, *ρ* has an associated Shannon entropy

$$S[\rho] = -\langle \ln \rho \rangle\_{\rho} \,. \tag{2}$$

.

By understanding *S* to be a measure of uncertainty quantifying our ignorance about the details of the system, the corresponding bias is minimised (compatibly with the prior data) by maximising *S* (under the set of constraints in Equation (1), plus the normalisation condition for *ρ*) [19]. The method of Lagrange multipliers then gives a generalised Gibbs distribution of the form,

$$\rho\_{\{\beta\_a\}} = \frac{1}{Z\_{\{\beta\_a\}}} \varepsilon^{-\sum\_a \beta\_a \mathcal{O}\_a} \tag{3}$$

where the partition function *<sup>Z</sup>*{*βa*} encodes all thermodynamic properties in principle, and is assumed to be convergent. This can be done analogously for a quantum system [20], giving a Gibbs density operator on a representation Hilbert space

$$\boldsymbol{\beta}\_{\{\beta\_a\}} = \frac{1}{Z\_{\{\beta\_a\}}} e^{-\sum\_a \beta\_a \mathcal{O}\_a} \,. \tag{4}$$

A generalised Gibbs state can thus be defined, characterised fully by a finite set of observables of interest O*<sup>a</sup>*, and their conjugate generalised "inverse temperatures" *β<sup>a</sup>*, which have entered formally as Lagrange multipliers. Given this class of equilibrium states, it should be evident that some thermodynamic quantities (e.g., generalised "energies" *Ua*) can be identified immediately. Aspects of a generalised thermodynamics are discussed in Section 2.4.

Finally, we note that the role of entropy is shown to be instrumental in defining (local6) equilibrium states: "...thus entropy becomes the primitive concept with which we work, more fundamental even than energy..." [19]. It is also interesting to notice that Bekenstein's arguments [6] can be observed to be influenced by Jaynes' information-theoretic insights surrounding entropy, and these same insights have now guided us in the issue of background independent statistical equilibrium.
