A New Thermodynamics, From Nuclei to Stars II

Equilibrium statistics of Hamiltonian systems is correctly described by the microcanonical ensemble. Classically this is the manifold of all points in the N-body phase space with the given total energy. Due to Boltzmann's principle, e^S=tr(\delta(E-H)), its geometrical size is related to the entropy S(E,N,...). This definition does not invoke any information theory, no thermodynamic limit, no extensivity, and no homogeneity assumption, as are needed in conventional (canonical) thermo-statistics. Therefore, it describes the equilibrium statistics of extensive as well of non-extensive systems. Due to this fact it is the general and fundamental definition of any classical equilibrium statistics. It can address nuclei and astrophysical objects as well. As these are not described by the conventional extensive Boltzmann-Gibbs thermodynamics, this is a mayor achievement of statistical mechanics. Moreover, all kind of phase transitions can be distinguished sharply and uniquely for even small systems. In contrast to the Yang-Lee singularities in Boltzmann-Gibbs canonical thermodynamics phase-separations are well treated.


Introduction
Classical Thermodynamics and the theory of phase transitions of homogeneous and large systems are some of the oldest and best established theories in physics. It may look strange to add anything new to this. Let me recapitulate what was told to us since > 100 years: • Thermodynamics addresses large homogeneous systems at equilibrium (in the thermodynamic limit N → ∞| N/V =ρ,homogeneous ).
• Phase transitions are the positive zeros of the grand-canonical partition sum Z(T, µ, V ) as function of e βµ (Yang-Lee-singularities). As the partition sum for a finite number of particles is always positive, zeros can only exist in the thermodynamic limit V | β,µ → ∞.
• Micro and canonical ensembles are equivalent.
• Thermodynamics works with intensive variables T, P, µ. Fluctuations can mostly be ignored.
• Unique Legendre mapping E ⇔ T .
• The heat capacity is always C > 0.
• Heat only flows from hot to cold (Clausius) • Second Law only in infinite systems when the Poincarré recurrence time becomes infinite (much larger than the age of the universe (Boltzmann)).
Under these constraint only a tiny part of the real world of equilibrium systems is addressed. The ubiquitous non-homogeneous systems: nuclei, clusters, polymer, soft matter (biological) systems, but even the largest, astrophysical systems are not covered. Even normal systems with short-range coupling at phase separations are non-homogeneous and can only be treated within conventional homogeneous thermodynamics (e.g. van-der-Waals theory) by bridging the unstable region of negative compressibility by an artificial Maxwell construction. Thus even the original goal, for which Thermodynamics was invented some 150 years ago, the description of steam engines is only artificially solved. There is no (grand-) canonical ensemble of phase separated and, consequently, non-homogeneous, configurations. This has a deep reason as I will discuss below: here the systems have a negative heat capacity (resp. susceptibility). This, however, is impossible in the (grand-)canonical theory where C ∝ (δE) 2

Boltzmann's principle
The Microcanonical ensemble is the ensemble (manifold) of all possible points in the 6N dimensional phase space at the prescribed sharp energy E: Thermodynamics addresses the whole ensemble. It is ruled by the topology of the geometrical size W (E, N, · · ·), Boltzmann's principle: is the most fundamental definition of the entropy S. Entropy and with it micro-canonical thermodynamics has therefore a pure mechanical, geometrical foundation. No information theoretical formulation is needed. Moreover, in contrast to the canonical entropy, S(E, N, ..) is everywhere single valued and multiple differentiable. There is no need for extensivity, no need of concavity, no need of additivity, and no need of the thermodynamic limit. This is a great advantage of the geometric foundation of equilibrium statistics over the conventional definition by the Boltzmann-Gibbs canonical theory. However, addressing entropy to finite eventually small systems we will face a new problem with Zermelo's objection against the monotonic rise of entropy, the Second Law, when the system approaches its equilibrium. This is discussed elsewhere [1,2]. A second difficulty is: Without the thermodynamic limit surface effects are not scaled away. However, this is important as the creation of inhomogeneities and interphase surfaces is on the very heart of phase transitions of first order.

Topological properties of S(E, · · ·)
The topology of the Hessian of S(E, · · ·), the determinant of curvatures of s(e, n) determines completely all kinds phase transitions. This is evidently so, because e S(E)−E/T is the weight of each energy in the canonical partition sum at given T . Consequently, at phase separation this has at least two maxima, the two phases. And in between two maxima there must be a minimum where the curvature of S(E) is positive. I.e. the positive curvature detects phase separation. This is of course also in the case of several conserved control parameters.
Of course for a finite system each of these maxima of S(E, · · ·) − E/T have a non-vanishing width. There are intrinsic fluctuations in each phase.

Unambiguous signal of phase transitions in a "Small" system
Nevertheless, the whole zoo of phase-transitions can be sharply seen and distinguished. This is here demonstrated for the Potts-gas model on a two dimensional lattice of finite size of 50 × 50 lattice points, c.f. fig.(1).  (2), of the 2-dim Potts-3 lattice gas with 50 * 50 lattice points, n is the number of particles per lattice point, e is the total energy per lattice point. The line (-2,1) to (0,0) is the ground-state energy of the lattice-gas as function of n. The most right curve is the locus of configurations with completely random spin-orientations (maximum entropy). The whole physics of the model plays between these two boundaries. At the dark-gray lines the Hessian is det = 0,this is the boundary of the region of phase separation (the triangle AP m B) with a negative Hessian (λ 1 > 0, λ 2 < 0). Here, we have Pseudo-Riemannian geometry. At the light-gray lines is a minimum of det(e, n) in the direction of the largest curvature (v λmax ·∇det = 0) and det = 0,these are lines of second order transition. In the triangle AP m C is the pure ordered (solid) phase (det > 0, λ 1 < 0). Above and right of the line CP m B is the pure disordered (gas) phase (det > 0, λ 1 < 0). The crossing P m of the boundary lines is a multi-critical point. It is also the critical end-point of the region of phase separation (det < 0, λ 1 > 0, λ 2 < 0). The lightgray region around the multi-critical point P m corresponds to a flat, horizontal region of det(e, n) ∼ 0 and consequently to a somewhat extended cylindrical region of s(e, n) and ∇λ 1 ∼ 0, details see [3,4]; C is the analytically known position of the critical point which the ordinary q = 3 Potts model (without vacancies)would have in the thermodynamic limit

Heat can flow from cold to hot
Clausius' first version of the second law is violated in regions of negative heat capacity. Taking energy out of such a system its temperature T = (dS/dE) −1 can rise whereas the temperature of the -originallyhotter recipient will drop. 4 Negative heat capacity as signal for a phase transition of first order.

Nuclear Physics
A very detailed illustration of the appearance of negative heat capacities in nuclear level densities is given by d.Agostino et al. [5].

Stars
The appearence of a negative heat capacity of equilibrized self gravitating systems is well known [6]. It was always considered as an absurd pitfall of thermodynamics. In our generalized theory this phenomenon turns out as the standard occurrence of pseudo-Riemannian geometry of S(E, L, N ) c.f. [7]. In the mixed phase the largest curvature λ 1 of S(E, L) is positive. Consequently the heat capacity or the correspondent susceptibility is negative. The new and important point of our finding is that within microcanonical thermodynamics this is a generic property of all phase transitions of first order, whether there is a shortor a long-range force that organizes the system. Self-gravitation leads to a non-extensive potential energy ∝ N 2 . No thermodynamic limit exists for E/N and no canonical treatment makes sense. At negative total energies these systems have a negative heat capacity. This was for a long time considered as an absurd situation within canonical statistical mechanics with its thermodynamic "limit". However, within our geometric theory this is just a simple example of the pseudo-Riemannian topology of the microcanonical entropy S(E, N ) provided that high densities with their non-gravitational physics, like nuclear hydrogen burning, are excluded. We treated the various phases of a self-gravitating cloud of particles as function of the total energy and angular momentum, c.f. the quoted paper. Clearly these are the most important constraint in astrophysics.

Conclusion
Entropy has a simple and elementary mechanical definition by the area e S(E,N,···) of the microcanonical ensemble in the 6N dim. phase space. Canonical ensembles are not equivalent to the micro-ensemble in the most interesting situations: 1. At phase-separation (−→heat engines !), one gets inhomgeneities, and a negative heat capacity or some other negative susceptibility, 2. Heat can flow from cold to hot.
3. Phase transitions can be localized sharply and unambiguously in small classical or quantum systems, there is no need for finite size scaling to identify the transition. 4. Also really large self-gravitating systems can be addressed.
Entropy rises during the approach to equilibrium, ∆S ≥ 0, also for small mixing systems. i.e. the Second Law is valid even for small systems [1,2]. With this geometric foundation thermo-statistics applies not only to extensive systems but also to non-extensive ones which have no thermodynamic limit. It is only by this extension of thermo-statistics that the real and deeper sources of thermodynamics become uncovered. It addresses a much larger world of phenomena. The old puzzle of the anomalous behavior of self-gravitational systems is resolved. The first time thermodynamics does justice to its original goal of treating properly the liquid-gas phase-separation, the motor of steam engines.