A New Thermodynamics from Nuclei to Stars
<p>Global phase diagram or contour plot of the curvature determinant (Hessian), eqn. (7), of the 2-dim Potts-3 lattice gas with 50 ∗ 50 lattice points, <span class="html-italic">n</span> is the number of particles per lattice point, <span class="html-italic">e</span> 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 <span class="html-italic">n</span>. 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 <span class="html-italic">AP<sub>m</sub>B</span>) with a negative Hessian (<span class="html-italic">λ</span><sub>1</sub> > 0, <span class="html-italic">λ</span><sub>2</sub> < 0). Here, we have Pseudo-Riemannian geometry. At the light-gray lines is a minimum of det(<span class="html-italic">e</span><span class="html-italic">, n</span>) in the direction of the largest curvature (v<sub><span class="html-italic">λ<sub>max</sub></span></sub> · ∇det = 0) and det = 0,these are lines of second order transition. In the triangle <span class="html-italic">AP<sub>m</sub>C</span> is the pure ordered (solid) phase (det > 0, <span class="html-italic">λ</span><sub>1</sub> < 0). Above and right of the line <span class="html-italic">CP<sub>m</sub>B</span> is the pure disordered (gas) phase (det > 0, <span class="html-italic">λ</span><sub>1</sub> < 0). The crossing <span class="html-italic">P<sub>m</sub></span> of the boundary lines is a multi-critical point. It is also the critical end-point of the region of phase separation (det < 0, <span class="html-italic">λ</span><sub>1</sub> > 0, <span class="html-italic">λ</span><sub>2</sub> < 0). The light-gray region around the multi-critical point <span class="html-italic">P<sub>m</sub></span> corresponds to a flat, horizontal region of det(<span class="html-italic">e</span>, <span class="html-italic">n</span>) ∼ 0 and consequently to a somewhat extended cylindrical region of <span class="html-italic">s</span>(<span class="html-italic">e</span>, <span class="html-italic">n</span>) and ∇<span class="html-italic">λ</span><sub>1</sub>∼ 0, details see [<a href="#B6-entropy-06-00158" class="html-bibr">6</a>, <a href="#B7-entropy-06-00158" class="html-bibr">7</a>]; <span class="html-italic">C</span> is the analytically known position of the critical point which the ordinary <span class="html-italic">q</span> = 3 Potts model (without vacancies) <span class="html-italic">would have in the thermodynamic limit</span></p> "> Figure 2
<p>MMMC [<a href="#B7-entropy-06-00158" class="html-bibr">7</a>] simulation of the entropy <span class="html-italic">s</span>(<span class="html-italic">e</span>) per atom (<span class="html-italic">e</span> in eV per atom) of a system of <span class="html-italic">N</span><sub>0</sub> = 1000 sodium atoms at an external pressure of 1 atm. At the energy <span class="html-italic">e</span> ≤ <span class="html-italic">e</span><sub>1</sub> the system is in the pure liquid phase and at <span class="html-italic">e</span> ≥ <span class="html-italic">e</span><sub>3</sub> in the pure gas phase, of course with fluctuations. The latent heat per atom is <span class="html-italic">q<sub>lat</sub></span> = <span class="html-italic">e</span><sub>3</sub> − <span class="html-italic">e</span><sub>1</sub>. <span class="underline">Attention:</span> the curve <span class="html-italic">s</span>(<span class="html-italic">e</span>) is artificially sheared by subtracting a linear function 25 + <span class="html-italic">e</span>∗11.5 in order to make the convex intruder visible. <span class="html-italic">s</span>(<span class="html-italic">e</span>) <span class="html-italic">is always a steep monotonic rising function</span>. We clearly see the global concave (downwards bending) nature of <span class="html-italic">s</span>(<span class="html-italic">e</span>) and its convex intruder. Its depth is the entropy loss due to additional correlations by the interfaces. It scales ∝ <span class="html-italic">N</span><sup>−1/3</sup> . From this one can calculate the surface tension per surface atom <span class="html-italic">σ<sub>surf</sub></span>/<span class="html-italic">T<sub>tr</sub></span> = ∆<span class="html-italic">s<sub>surf</sub></span> ∗ <span class="html-italic">N</span><sub>0</sub>/<span class="html-italic">N<sub>surf</sub></span>. The double tangent (Gibbs construction) is the concave hull of <span class="html-italic">s</span>(<span class="html-italic">e</span>). Its derivative gives the Maxwell line in the caloric curve <span class="html-italic">T</span>(<span class="html-italic">e</span>) at <span class="html-italic">T<sub>tr</sub></span>. In the thermodynamic limit the intruder would disappear and <span class="html-italic">s</span>(<span class="html-italic">e</span>) would approach the double tangent from below. Nevertheless, even there, the probability ∝ <span class="html-italic">e<sup>Ns</sup></span> of configurations with phase-separations are suppressed by the (infinitesimal small) factor <math display="inline"> <semantics> <mrow> <msup> <mi>e</mi> <mrow> <mo>−</mo> <msup> <mi>N</mi> <mrow> <mn>2</mn> <mo>/</mo> <mn>3</mn> </mrow> </msup> </mrow> </msup> </mrow> </semantics> </math> relative to the pure phases and the distribution remains <span class="html-italic">strictly</span> <span class="html-italic">bimodal</span><span class="html-italic">in the canonical ensemble</span>. The region <span class="html-italic">e</span><sub>1</sub> < <span class="html-italic">e</span> < <span class="html-italic">e</span><sub>3</sub> of phase separation gets lost.</p> "> Figure 3
<p>Potts model, (<span class="html-italic">q</span> = 10) in the region of phase separation. At <span class="html-italic">e</span><sub>1</sub> the system is in the pure ordered phase, at <span class="html-italic">e</span><sub>3</sub> in the pure disordered phase. A little above <span class="html-italic">e</span><sub>1</sub> the temperature <span class="html-italic">T</span> = 1<span class="html-italic">/β</span> is higher than a little below <span class="html-italic">e</span><sub>3</sub>. Combining two parts of the system: one at the energy <span class="html-italic">e</span><sub>1</sub> + <span class="html-italic">δe</span> and at the temperature <span class="html-italic">T</span><sub>1</sub>, the other at the energy <span class="html-italic">e</span><sub>3</sub> − <span class="html-italic">δe</span> and at the temperature <span class="html-italic">T</span><sub>3</sub><span class="html-italic">< T</span><sub>1</sub> will equilibrize with a rise of its entropy, a drop of <span class="html-italic">T</span><sub>1</sub> (cooling) and an energy flow (heat) from 3 → 1: i.e.: Heat flows from cold to hot! Clausius formulation of the second law is violated. Evidently, this is not any peculiarity of gravitating systems! This is a generic situation within classical thermodynamics even of systems with short-range coupling and <span class="html-italic">has nothing to do with long range interaction.</span></p> "> Figure 4
<p>Experimental excitation energy per nucleon <span class="html-italic">e</span>* versus apparent temperature <span class="html-italic">T<sub>app</sub></span> for backward <span class="html-italic">p</span>, <span class="html-italic">d</span>, <span class="html-italic">t</span> and <span class="html-italic">α</span> together with heavy evaporation residues out of incomplete fusion of 701 Mev <sup>28</sup>Si+<sup>100</sup>Mo. The dotted curves give the Fermi-gas caloric curves for the level-density parameter a = 6 to 12. (Chbihi et al. Eur.Phys.J. A 1999)</p> "> Figure 5
<p>Cluster fragmentation</p> "> Figure 6
<p>Phases and Phase-Separation in Rotating, Self-Gravitating Systems, Physical Review Letters–July 15, 2002, cover-page, by (Votyakov, Hidmi, De Martino, Gross)</p> "> Figure 7
<p>Microcanonical phase-diagram of a cloud of self-gravitating and rotating system as function of the energy and angular-momentum. Outside the dashed boundaries only some singular points were calculated. In the mixed phase the largest curvature <span class="html-italic">λ</span><sub>1</sub> of <span class="html-italic">S</span>(<span class="html-italic">E</span>, <span class="html-italic">L</span>) is positive. Consequently the heat capacity or the correspondent susceptibility is negative. This is of course well known in astrophysics. However, the new and important point of our finding is that within microcanonical thermodynamics this is <span class="html-italic">a generic property of all phase transitions of first order, independently of whether there is a short- or a long-range force that organizes the system</span>.</p> "> Figure 8
<p>The compact set <math display="inline"> <semantics> <mrow> <mi mathvariant="bold-italic">M</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="bold-italic">t</mi> <mn mathvariant="bold">0</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math>, left side, develops into an increasingly folded “spaghetti”-like distribution in phase-space with rising time <span class="html-italic">t</span>, Gibbs’ ”ink-lines”. The right figure shows only the early form of the distribution. At much larger times it will become more and more fractal and finally dense in the new phase space. The grid illustrates the boxes of the box-counting method. All boxes which overlap with <math display="inline"> <semantics> <mrow> <mi mathvariant="bold-italic">M</mi> <mrow> <mo>(</mo> <mi mathvariant="bold-italic">t</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> are counted in <b><span class="html-italic">N<sub>δ</sub></span></b> in eq.(20)</p> ">
Abstract
:1 Introduction
- 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 eβµ (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. 1
- thermodynamics works with intensive variables T, P, µ.
- Unique Legendre mapping T → E.
- 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)).
2 Boltzmann’s principle
3 Topological properties of S(E, ⋯)
3.1 Unambiguous signal of phase transitions in a ”Small” system [5]
3.2 Systematic of phase transitions in the micro-canonical ensemble without invoking the thermodynamic limit
- A single stable phase of course with some intrinsic fluctuations (width) by a negative largest curvature λ1 < 0. Here s(e, n) is concave (downwards bending) in both directions. Then there is a one to one mapping of the canonical ↔the micro-ensemble.
- A transition of first order with phase separation and surface tension is indicated by λ1(e, n) > 0. s(e, n) has a convex intruder (upwards bending) in the direction v1 of the largest curvature ≥ 0 which can be identified with the order parameter [7]. Three solutions ofIn the thermodynamic limit the whole region {o1, o3} is mapped into a single point in the canonical ensemble which is consequently non-local in o. I.e. if the curvature of S(E, N ) is λ1 ≥ 0 both ensembles are not equivalent even in the limit.
- A continuous (“second order”) transition with vanishing surface tension, where two neighboring phases become indistinguishable. This is indicated in figure (1) by a line with λ1 = 0 and extremum of λ1 in the direction of order parameter vλ=0 · ∇λ1 = 0. These are the catastrophes of the Laplace transform E → T .
3.3 CURVATURE
N0 | 200 | 1000 | 3000 | bulk | |
Na | Ttr [K] | 940 | 990 | 1095 | 1156 |
qlat [eV ] | 0.82 | 0.91 | 0.94 | 0.923 | |
sboil | 10.1 | 10.7 | 9.9 | 9.267 | |
∆ssurf | 0.55 | 0.56 | 0.44 | ||
Nsurf | 39.94 | 98.53 | 186.6 | ∞ | |
σ/Ttr | 2.75 | 5.68 | 7.07 | 7.41 |
3.4 Heat can flow from cold to hot
4 Negative heat capacity as signal for a phase transition of first order.
4.1 Nuclear Physics
4.2 Atomic clusters
4.3 Stars
5 The second law, microcanonically
5.1 Measuring a macroscopic observable, the “ EPS-formulation ”
5.2 Fractal distributions in phase space, second law
5.3 Conclusion of the discussion of the second law
6 Final conclusion
- at phase-separation (→heat engines !), one gets inhomgeneities, and a negative heat ca-pacity or some other negative susceptibility,
- Heat can flow from cold to hot.
- 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.
- also really large self-gravitating systems can now be addressed.
7 Appendix
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- 1How does one normally prove this?: The general link between the microcanonical probability eS(E,N) and the grand-canonical partition function Z(T, µ) for extensive systems is by the Laplace transform:
- 2In this paper we denote ensembles or manifold in phase space by calligraphic letters like
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Gross, D.H.E. A New Thermodynamics from Nuclei to Stars. Entropy 2004, 6, 158-179. https://doi.org/10.3390/e6010158
Gross DHE. A New Thermodynamics from Nuclei to Stars. Entropy. 2004; 6(1):158-179. https://doi.org/10.3390/e6010158
Chicago/Turabian StyleGross, Dieter H.E. 2004. "A New Thermodynamics from Nuclei to Stars" Entropy 6, no. 1: 158-179. https://doi.org/10.3390/e6010158