Negative specific heat of a magnetically self-confined plasma torus

Footnotes

• ↵ † To whom correspondence should be addressed. E-mail: miki{at}math.rutgers.edu.

• ↵ § Because any probability density maximizes the entropy relative to itself, a stationary plasma torus is necessarily a maximum-entropy configuration under some constraints. What makes the maximum-entropy proposal nonempty is the insistence on only a few global natural dynamical constraints.

• ↵ ¶ The first qualitative predictions based on statistical mechanics of the Hamiltonian system of N point vortices were made in ref. 11. The quantitative evaluation began with ref. 12; impressive agreement with simulated flows is reported in ref. 13. Its mathematical rigorous foundations are almost complete by now, the latest word being ref. 14; see ref. 15 for a review. More recently a formulation based directly on continuum vorticity has gained much ground; see ref. 16 for a state-of-the-art report.

• ↵ ‖ In the guiding center approximation, the dynamics of this plasma system is identical to that of N point vortices (17). Statistical mechanics in the co-rotating frame predicts that at high enough effective energies the nonlinear m = 1 diocotron mode has higher entropy than any other configuration with the same energy and angular momentum (18, 19), which is in accordance with remarkable real experiments (20).

• ↵ ** Ref. 21 (pp. 60–63) explains why a homogeneous piece of “everyday matter” must have positive specific heat. See p. 62 of ref. 21 for why those arguments do not rule out negative specific heat in an isolated gravitating system. Indeed, the virial and equipartition theorems imply that in a spherical equilibrium system the energy is distributed −2:1 between gravitational and kinetic. A decrease in total energy E of a gravitational equilibrium gas ball will increase its thermal energy. Such a system grows hotter while losing energy through, say, radiation. Negative specific heat in self-gravitational perfect gases is evaluated quantitatively already in ref. 22 and is discussed further in refs. 23 and 24; however, none of these configurations with negative specific heat is thermodynamically stable, although some are metastable. Thermodynamically stable self-gravitating configurations with negative specific heat can occur when the Newtonian −1/r singularity is stabilized as in quantum mechanics (25), classical hard-balls systems (26), or a model of concentric, self-gravitating, mass shells (27, 28). In this last work the dynamical stability of energetically isolated equilibrium states with negative specific heat, predicted by theory, was also verified by dynamical simulations.

• ↵ ‡‡ Recently, the existence and importance of negative specific heat was also reported for the diocotron mode of the guiding center plasma (18, 19) alias point vortex gas and for certain vorticity structures in geostrophic flows (16, 32). However, very different from the gravothermal-type negative specific heat that we report here to be a characteristic also of the magnetically self-confined plasma torus, the negative specific heat of these quasiparticle systems does not couple to the thermal motion of the underlying physical particle systems, which is evident from the fact that these quasiparticle systems also exhibit negative temperature (11).

• ↵ †† The negative sign in front of the magnetic energy in W is due to the canonical constraint of prescribed electric current; compare with the negative sign in front of the centrifugal contribution to the kinetic energy of a rotating thermal system in the co-rotating frame (see Landau and Lifshitz, ref. 21, pp. 71–73). Incidentally, those very centrifugal contributions to W are negligible in our plasma and have been omitted. Moreover, the toroidal field B _T does not show, because we consider only axisymmetric configurations with toroidal current density.

• ↵ §§ These are quite nontrivial facts. In particular, all this is not true if we relax the condition of axisymmetry.

• Copyright © 2003, The National Academy of Sciences