Magnetic energy per particle in constant current density

We assume a constant current density in a homogeneous one-component plasma of infinite extent and calculate the resulting magnetic energy per particle. Our starting point is the conserved approximately relativistic (Darwin) energy for a system of electromagnetically interacting particles that arises from the neglect of radiation. For the idealized model of a homogeneous one-component plasma the energy only depends on the particle canonical momenta and the vector potential. The vector potential is then calculated in terms of the canonical momenta using recent theoretical advances and the plasma Hamiltonian is obtained. The result can be understood either as due to the energy lowering caused by the attraction of parallel currents or, alternatively, as due to the inductive inertia associated with the flow of net current. Copyright c © EPLA, 2008 Our theoretical understanding of matter is largely based on the Coulomb interaction between charged particles. For small systems one can usually assume that the effects of the magnetic corrections are secondary but for larger systems this it not the case. Including magnetic interaction into the theories, however, has met with considerable difficulties, from the 1939 “magnetische Katastrophe” of Welker [1] to the 1999 rigorous proof of the instability of matter with magnetic interaction by Griesemer and Tix [2]. Their result is valid whether the magnetic interaction is mediated by the (Darwin-) Breit potential or via a quantized radiation field. In both these cases it is the attraction of parallel currents that causes the problem. This attraction, which is so fundamental that it is used in the definition of the ampere, the unit of electric current in the SI-system, indicates a long-range energy lowering due to correlation of currents that seems to diverge in large systems. Laboratory and astrophysical plasmas are observed to harbor intense currents and magnetic fields [3], but in plasma physics it is usually assumed that this is due to non-equilibrium, and that the equilibrium plasma is described by the traditional Maxwell-Boltzmann distribution (see, e.g., Burm [4]). This is clearly at odds with the above findings of an instability of the energy minimum ground state to parallel current generation. Alastuey (a)E-mail: [email protected]; URL: http://www.mech.kth.se/∼ hanno/ and Appel [5] claim that inclusion of the quantized radiation field removes the instability problem, in direct contradiction with the findings of Griesemer and Tix [2]. Most plasma physicists do not seem to be aware of the problem even if there certainly has been a fair amount of interest in energy extremizing states and self-organization, see, e.g., Woltjer [6], Taylor [7]. Here we will show that within a simple standard model, based on classical electrodynamics and relativistic Hamiltonian mechanics, the energy of a plasma is considerably reduced when the canonical momenta are correlated (parallel) and thus that conclusions drawn from the traditional non-relativistic Maxwell-Boltzmann distribution of non-interacting particles, or particles interacting only via a Debye screened Coulomb potential, cannot be trusted. The model, which neglects radiation, gives a quantitative estimate of the energy reduction, but does not lead to any unphysical divergence. Let us start from the following expression for the energy of a system of classical charged particles and electromagnetic fields: