Entropy
The reason why the classical laws of motion do not address thermodynamic phenomena is that they concern one-dimensional trajectories, while thermodynamics concerns three-dimensional volumes. For example, Einstein in his paper “Concerning an Heuristic Point of View Toward the Emission and Transformation of Light”, section 5, presents an alternative version of statistical mechanics where entropy is defined in terms of physical volume v. On page 10, substituting k for R/N gives S – S0 = kn ln (v/v0) or S = kn ln v. If we imagine the case of n particles initially trapped in a spherical balloon suspended in free space, we can calculate Einstein’s entropy as a function of time. The initial volume of the balloon is given by v0 = 4πr03/3, where r0 is the radius. If we assume that the particles have a maximum speed of s but are oriented randomly, then when the balloon is popped at time t0 = 0, the radius of the sphere containing all of the particles will be r = r0 + st and the volume containing the will be v = 4πr3/3. Taking the simple case of the big bang where r0 = 0 and r = st and setting s = c, the speed of light, one gets S = kn (ln v) = kn ln [(4/3)π(ct)3], where n is proportional to the conserved mass/energy of the universe. Differentiating this gives dS/dt = 3kn/t or in terms of Boltzmann’s H, dH/dt = – 3n/t.
We see from this that dH/dt ≤ 0 since t ≥ 0. However, Boltzmann pulls his H out of the air, while here it is derived from first principles. The reversibility argument is the same as previously made for the spatial dispersion of particles. In fact, the referenced quote from Thomson (1874) relates to velocity and energy as well as physical space.
Boltzmann cosmology implies that the matter present at the big bang expands into infinite empty space. Einstein cosmology implies that space itself is expanding and that no space exists “prior” to the big bang. This corresponds with Hawking’s statement that “At the big bang itself, the universe is thought to have had zero size”. [A Brief History of Time, p. 117] Just as we have no evidence of time before the big bang, we have no evidence of space beyond the borders of the universe.
Energy Distribution
In footnote 6 on page 10, Einstein suggests a simple way to derive the energy distribution without resorting to the Lagrange multiplier. This approach involves the following steps for deriving the distribution of the energy states over the volume of an ideal gas.
pV = NkT for an ideal gas at equilibrium.
dU = − p dV for an adiabatic system with internal energy U.
dU = − NkT dV/V from combining the above.
dV/V = − dU/NkT rearranged.
ln V + C = − U/NkT integrated.
V = exp[− U/NkT + C] ∝ exp[− U/NkT]
Ui = Ni εi where εi is the energy of the ith energy state
and U = Σ Ui.
Vi ∝ exp[− εi /kT] where Vi is the volume containing the Ni particles.
Assuming that at equilibrium the particles are distributed uniformly over the volume V = Σ Vi results in the probability distribution
pi = Ni /N = Vi /V = (1/Z) exp[− εi /kT],
where Z = Σ exp[− εi /kT] over all i.
The a priori assumption of equiprobability over physical space is more plausible than Boltzmann’s a priori assumption of equiprobability over energy states and avoids the associated logical contradiction between the “fundamental postulate of statistical mechanics” and the geometrical result of the energy distribution.
References:
Einstein Heuristic Point of View
Revised 10/23/22