Phase-space energy of charged particles with negligible radiation: Proof of spontaneous formation of magnetic structures and new effective forces Hanno Esse ´ n Department of Mechanics, Royal Institute of Technology, S-100 44 Stockholm, Sweden ~Received 12 March 1997! The phase-space energy of a system of charged particles in the negligible radiation ~Darwin! nonrelativistic limit is derived. The usefulness of the second-order approximation to this Hamiltonian, previously found by the present author, is discussed and given a stronger theoretical foundation. As a result a typical length scale for magnetic structures is found. The virial theorem is then applied to the second-order Hamiltonian. Together with the assumption that magnetism is a first-order perturbation it proves that magnetic interaction lowers the energy of a plasma. This proves that net currents must flow in a plasma and a simple estimate indicates that the spontaneously forming magnetic structures are resistant to thermal disruption. The effective one-particle Hamiltonian implied by the second-order Hamiltonian is calculated. It predicts a new effective many-body force that accelerates charged particles near a large plasma. It is conjectured that this effective force could give a simpler explanation for stellar wind and other large scale plasma phenomena. @S1063-651X~97!01211-7# PACS number~s!: 52.25.Kn, 51.60.1a, 41.20.Bt, 97.10.Me I. ON DARWIN’S APPROACH TO ELECTROMAGNETISM We live in a world built from charged particles. We know the exact theory needed to predict their behavior: Maxwell’s equations and the Lorentz force law. Yet, when there are macroscopic numbers of free charged particles, as in a plasma or metal, the resulting coupled equations for particles and fields become so complicated that approximations are necessary. The standard approximation for plasmas is mag- netohydrodynamics. In spite of the approximations, it has been very difficult to achieve a useful understanding of plasma behavior using this theory. Here we will take a radi- cally different approach. We will study the finite degree of freedom many-body system that best approximates the real system, using analytical mechanics. As long as radiation can be neglected, a system of charged particles has a conserved energy. Usually the corre- sponding Hamiltonian is taken to contain the Coulomb, or electrostatic, interaction energy only. In 1920 Darwin @1# re- alized that magnetic effects can be included in such a de- scription up to terms that are proportional to ( v / c ) 2 . The Darwin approach leads first to a Lagrangian @2–6#. Darwin found a first-order approximation to the corresponding Hamiltonian @1,7–9# but, for a long time, no exact explicit expression for it was known. The first-order Hamiltonian, which gives excellent results for few-body systems ~see @10# for a careful treatment of two-body systems! is, however, not correct for macroscopic systems. The crucial difference be- tween these is that in few-body systems magnetic effects are always relativistic whereas in macroscopic systems this is not normally the case @3#. In 1968 Trubnikov and Kosachev @11# found a formally exact expression for the Darwin Hamiltonian in terms of a series expansion for the canonical momentum. This expan- sion, however, would not converge for the case of a macro- scopic plasma. The present author reconsidered the problem and found a different expansion for the dependence of the vector potential on the canonical momenta. This led to a closed expression for a second-order term and a physically reasonable second-order Hamiltonian @12#. The question of convergence was not completely resolved so we return to it here. The first part of this paper gives strong arguments that the second-order Darwin Hamiltonian, formula ~19! or ~21!, is qualitatively correct and physically useful for macroscopic systems. The Darwin approximation to the equations of motion for charged particles has been used before in plasma physics with considerable success @13–15#. The Hamiltonian, on the other hand, has not been studied much in spite of the great importance of the Hamiltonian formalism in statistical phys- ics and the great simplicity and generality of energy consid- erations. The reason for this has been uselessness of the first- order Hamiltonian and the unknown, or intractable, form of the exact Hamiltonian. The new second-order term that makes the approximate Hamiltonian qualitatively correct is therefore quite important. This second-order Darwin Hamiltonian has far reaching consequences for the nature of the thermal equilibrium of systems containing mobile charged particles with kinetic en- ergy. As discussed in previous publications by the present author the magnetic interaction can be expected to play an important role in low-temperature superconductivity @16,12#. Here we calculate, using the virial theorem, the time aver- ages of different contributions to the energy in this Hamil- tonian. We find that the magnetic energy really does lower the energy of a plasma. This, of course, means that net cur- rents must flow in the plasma; otherwise there would not be any magnetic effects. We also estimate the size and energy of the spontaneous magnetic structures and find that they should survive thermal fluctuations. Finally we study the effective one-particle Hamiltonian that arises from the second-order Darwin Hamiltonian when one considers the motion of one of the particles assuming given positions and momenta of all other particles. This is seen to lead to the usual result for the case of a few-particle system. When macroscopic numbers contribute, on the other PHYSICAL REVIEW E NOVEMBER 1997 VOLUME 56, NUMBER 5 56 1063-651X/97/56~5!/5858~8!/$10.00 5858 © 1997 The American Physical Society