Retarded Electromagnetic Interaction and Dynamic Foundation of Classical Statistical Mechanics

It is provided in the paper that the non-conservative dissipative force and asymmetry of time reversal can be naturally introduced into classical statistical mechanics after retarded electromagnetic interaction between charged micro-particles is considered. In this way, the rational dynamic foundation for classical statistical physics can established and the revised Liouville’s equation is obtained. The micro-canonical ensemble, the canonical ensemble, the distribution of near-independent subsystem, the distribution law of the Maxwell-Boltzmann and the Maxwell distribution of velocities are achieved directly from the Liouville’s equation without using the hypothesis of equal probability. The micro-canonical ensemble is considered unsuitable as the foundation of equivalent state theory again, for most of equivalent states of isolated systems are not the states with equal probabilities actually. The reversibility paradoxes in the processes of non-equivalent evolutions of macro-systems can be eliminated completely, and the united description of statistical mechanics of equivalent and non-equivalent states is reached. The revised BBKGY series equations and hydromechanics equations are reduced, the non-equivalent entropy of general systems is defined and the principle of entropy increment of the non-equivalent entropy is proved at last. 1. The fundamental problems existing in classical statistical physics Though classical statistical mechanics has been highly developed, its foundation has not yet been built up well up to now . The first problem is about the rationality of the equal probability hypothesis or the micro-canonical ensemble hypothesiswhich is used as the foundation of equilibrium theory now. The hypothesis had got much criticism since it was put forward. In order to provide the hypothesis a rational base, Boltzmann raised the ergodic theory, proving that as long as a system was ergodic, the hypothesis of equal probability would be tenable. However, the study shows that the evolutions of systems can’t be ergodic generally . So the hypothesis of equal probability can only be regarded as a useful work principle without strict proof. As for the non-equilibrium statistical systems, we have no united and perfect theory at present. We do not know how a system to transform from a non-equilibrium state to an equilibrium one at present and how to define the non-equivalent entropy of general systems. Besides, there still exists a so-called reversibility paradox in the theory. Though lots of researches have been done, the really rational solution still remains to be explored. The reversibility paradox has been a long-standing problem. There exist two forms of the reversibility paradox at present. The first is the so-called Poincanre’s recurrence. It was put forward by Zermelo in 1896 based on a theorem provided by Poincanre in 1890 . According to this form, a conservative system in a limited space would return to the infinitely nearing neighbor reign of its initial state. The basic ideal of the proof is described as follows. For a conservative system, the Liouville’s theorem is tenable, so the volume of phase space is unchanged in the evolution process. Because the volume of phase space is limited, the system should be recurrent after a long enough time’s evolution. That is to say, the process is reversible. The second form was put forward by Loschmidt in 1876. Loschmidt thought that any micro-dynamic motion equation did not change under time reversal, or the motion of any single micro-particle was reversible, so after the velocities of all particles in a macro-system are reversed at the same time, the system would evolve along the completely opposite direction. Therefore, the process would be reversible. However, in the evolution processes of isolated macro-systems, what can be observed is that the processes are always irreversible. Therefore, there exists the so-called reversibility paradox. Though a large numbers of explanations have been given up to now, none of them are satisfying. 2. The Lorents retarded force and the basic hypothesis of classical statistical physics It is well-known that there exist two kinds of motion equations, one is for single particles, the other is for a statistical systems composed of a larger numbers of particles. The so-called micro-equation of motion that Loschmidt mentioned about in his time was actually the Newtonian equation md~r/dt = ~ F . In that time quantum mechanics has not been found. Whether the Newtonian equation can keep unchanged under time reversal depends on the form of force ~ F . Only when ~ F is conservative, it dose. In general situations when ~ F is relative to time t and momentum ~ p, the Newtonian equation can’t keep unchanged under time reversal in general. In the common statistical systems composed of charged micro-particles, the interaction forces between micro-particles are electromagnetic forces. In the systems composed of neutral atoms and molecules, atoms and molecules can be regarded as electromagnetic dipole moments and quadrupole moments for their deformations caused by interactions and interaction between them can considered as one between electromagnetic dipole moments and quadrupole moments. In the paper, we only discuss interaction between charged particles, but the principle is the same for neutral atoms and molecules. If the retarded interaction is not considered, the Lorentz force between two particles with charges q and q as well as velocities ~v and ~v is ~ F = qq~r r3 + qq~v × (~v × ~r) c2r2 (1) In the current statistical physics, we use the formula to describe interactions between micro-particles. Because the force is unchanged when ~v → −~v and ~v → −~v under time reversal, it is impossible for us to introduce the irreversibility of time reversal into the theory to solve the problems in non-equivalent statistical mechanics. However, it should be noted that according to special relativity, instantaneous interaction does not exist. Electromagnetic interaction propagates in the speed of light. So retarded interaction should be considered between micro-particles in statistical systems. It is proved below that after retarded interaction is considered, the Lorentz forces can not keep unchanged under time reversal again and a rational dynamic foundation can be established for classical statistical mechanics and the reversibility paradox can resolved well. Let t, ~r, ~v and ~a represent retarded time, coordinate, velocity and acceleration, t, ~r, ~v and represent nonretarded time, coordinate, velocity and acceleration. A particle with charge qj , velocity ~v ′ j and acceleration ~aj at space point ~r ′ j(t ) and time t would cause retarded potentials as follows at space point ~ri(t) and time t φij(~ri, t) = qj (1− ~ν j · ~nij ′c)r ′ ij ~ Aij(~ri, t) = qj~ν ′ j c(1− ~ν j · ~nij ′c)r ′ ij (2) In the formula ~r ij(t, t ) = ~r i(t)−~r ′ j(t ), r ij =| ~r ′ ij | ~nij = ~r ′ ij/r ′ ij . Let ~a ′ j represent j particle’s acceleration at time t, as well as v jn = ~v ′ j · ~njn a ′ jn = ~a ′ j · ~nij the intensities of electromagnetic fields caused by j particle at the space point ~ri at time t are ~ Eij(~ri, t) = q(1 − v j c )(~nij − ~v′ j c ) (1− v jn c ) 3r′2 ij + q~nij × [(~nij − ~v′ j c )× ~a ′ j] c2(1− v jn c ) 3r ij ~ Bij(~ri, t) = ~nij × ~ Eij (3) We call the forces as the retarded Lorentz forces in which retarded interaction has been considered. Suppose there are N particles in the system. Suppose the i particle with charge qi at space point ~ri(t) at time t moves in velocity ~vi, the retarded Lorentz force acted on the i particle caused by the j particle is ~ FRij = qiqj(1− v ′ jn/c ) (1− v jn/c) 3r′2 ij [~nij(1 − ~vi · ~v ′ j c2 )− ~v j c (1− vin c )] + qiqj c2(1− v jn/c) 3r ij {~nij [a ′ jn(1− ~vi · ~v ′ j c2 )− ~vi · ~a ′ j c (1 − v jn c )] − ~v ja ′ jn c (1− vin c )− ~a′j(1 − vin c )(1− v jn c )} (4) In the formula, vin = ~nij ·~vi, v ′ jn = ~nij ·~v ′ j , a ′ jn = ~n ′ ij ·~a ′ j . In order to do rational approximate calculation, let’s accumulate the magnitude order of acceleration. According to formula a = v/r, a/c has the magnitude order of 1/r. Suppose hydrogen atom can be regarded as a harmonic oscillator with amplitude b and angle frequency ω, so the acceleration of oscillator is bω. The magnitude order of energy of hydrogen atom is E = hc/λ = mv = mbω. Suppose the wavelength of photons emitted by hydrogen atom is λ = 4 × 10M , it can be calculated that a/c = 1/r ≃ 60. But the magnitude order of distance between atoms is r ≃ 10M , 1/r ≃ 10. According to Eq.(4), the first item of the retarded Lorentz force directs ratio to 1/r. So for the neighbor interaction, we have a/cr = 1/rr = 6× 10/r2 << 1/r, the item containing acceleration can be omitted. For the distance interaction, a/c ≥ 1/r, the items containing acceleration can not be omitted On the other hand, according to the Maxwell distribution of velocity, the average speed of hydrogen atoms is v̄ = √ 8kT/πm. Under common temperature T = 300K, we have v̄ = 6.3 × 10M , v̄/c = 2.1 × 10. So the item containing v/c is much bigger than the item containing acceleration, even the items containing v/c are bigger than that containing acceleration in near neighbor interaction. So we should retain the items containing v/c, v/c and va/c, but omitted high order items, then write Eq.(4) as ~ FRij = qiqj r′2 ij [~nij(1 + 3v jn c + 5v jn c2 − v j c2 − ~vi · ~v ′ j c2 )− ~v j c (1− vin c + 3v jn c )] + qiqj c2r ij {~nij [a ′ ij(1 + 3v jn c )− ~vi · ~a ′ j c ]− ~v ja ′ jn c − ~a′j(1− vin c + 2v jn c )} (5) For the convenience of discussion later, we write ~ F ′ Rij = ~ F ′ 0ij + ~ F ′ ij , ~ F ′ 0ij = qiqj~nij/r ′2 ij represent conservative part and ~ F ′ ij represents non-conservative part. The total force acted on the i-particle caused by the other particles is ~ F ′ Ri = ∑ ~ F ′ 0ij + ∑ ~ F ′ ij = ~ F ′ 0i + ~ F ′ i . For the macro-systems composed of large numbers of neutral atoms and molecular, we can consider atoms and molecular as electromagnetic dipole moments and quadrupoles and obtain the retarded Lorents forces in the same way. On the other hand, according to classical electromagnetic theory, there exists the radiation damping forces ~ G acted on the accelerated particles. The radiation damping forces are relative to the acceleration of acceleration. Suppose l is the dimension of the particles, particle’s speed v << c, acceleration a << c/l, the acceleration of acceleration ȧ << c/l, when the distribution of particle’s charge is with spherical symmetry, we have ~ G = κ~̇a, κ = 2q/3c. In classical electromagnetic theory, the Hamiltonian equations of motions can be described by using the space coordinates and common momentums ~pi, or by using the spaces coordinates and canonical momentums ~ pzi with relation ~pzi = ~ pi + qi ~ Ai/c. Both are equivalent. In statistical mechanics, it is more convenient to use common momentums. The accelerations and the accelerations of accelerations of particles should be regarded as the function of coordinate, momentum and time. On the other hand, the acceleration of the -particle is caused by other particle’s interactions, and it is relative to the retarded speeds and distances, i,e., ~ai = ~a ′ i(~ri, ~ pi, ~rj , ~ pj), ~̇a ′ i = ~̇a ′ i(~ri, ~pi, ~r ′ j , ~ p ′ j). So the total Hamiltonian of the non-conservative system composed of N charged particles can be written as H = H0 +H , in which H0 is the conservative Hamiltonian H0 = ∑