Microscopic Foundations of Thermodynamics and Generalized Statistical Ensembles Page: 47
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U)
0.3
O 0.25-
E 0.2-
U)
0.15-
0.1--
0.05
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-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15
energy
FIGURE 5.4. Temperature versus Energy for a particle of mass rn 1
in a Lennard-Jones box of size V = 4 > Vc. The curve displays a region
of negative heat capacity. At the critical energy the temperature goes
to zero.
5.3. Discussion
The example provided in the previous section is perhaps too simple to be of
interest to any specific physical problem. Nonetheless it illustrates qualitatively the
mechanism of microcanonical phase transition as captured by the volume entropy.
Such phase transitions are associated with the crossing of separatrix trajectories, for
which the dynamics of the system has no finite time scale. The figures show neatly
that at the separatrix energy the entropy has a discontinuous jump, the temperature
goes to zero, and for energies below the critical energy we have a region of negative
heat capacity. These are not special features of the system studied.1 Whenever a
separatrix is crossed there is a sudden open-up of a larger portion of phase space
1A similar behavior has been observed in a chain of particles interacting via a q4 potential
studied with volume entropy [54] and has been predicted for the pendulum [50].47
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Campisi, Michele. Microscopic Foundations of Thermodynamics and Generalized Statistical Ensembles, dissertation, May 2008; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc6128/m1/57/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .