A statistical analysis of Lorentz invariance in entropy

After relativity was introduced by Einstein in 1905, several attempts were made to reformulate the other fields of physics to make them consistent with the postulates of Relativity. However, relativistic reformulation remains a bone of contention even today after over a hundred years since Einstein's epoch making papers. This is mainly due to seemingly conflicting yet consistent ways of defining the macroscopic variables to be Lorentz invariant. Here, it will be shown from statistical origins that the current definition of entropy is untenable in relativistic scenarios.


INTRODUCTION
There are various ways to go about transforming the first and second laws of thermodynamics.One approach (Plank-Einstein) is to consider the invariance of pressure and obtain (Callen and Horwitz, 1971): From standard length contraction formulation.We are going to use standard notation implying: P is the pressure, U is the internal energy, Q is the Heat content, V is the volume and T is the temperature.In the term, v is the relative velocity of the frame while c is the speed of light.Also, the subscript 0 has been used to indicate the macroscopic variables in the rest frame).And therefore from standard Charles' Law: Also extending using the first law ( : The second approach (Ott) is to consider the conservation of momentum before and after the thermodynamic process to get inverse of the above relations (Dunkel et al., 2009).
Simply by considering the energy transferred as seen from a moving frame (Rothenstein and Zaharie, 2003): And therefore by the invariance of entropy This will also lead to: Note, that since the first law has been used to obtain these relations, none of them violate it or provide any inconsistencies to the definition of entropy which is considered invariant in both the cases.The problem of course arises when we choose to consider the results of an observation.Thus here, we choose to look at a more fundamental way by looking at the statistical origins of entropy.

Let's begin with Boltzmann's ubiquitous equation:
There was a great deal of controversy, when this equation was introduced by Boltzmann a century ago (Campisi and Kobe, 2010).Succinctly Einstein had said, the equation lacks proper theoretical basis and the concept of microstates is heuristic at best.However, nevertheless it does simplify quite a lot of scenarios and a relativistic reformulation is much needed.

Modified H-Theorem
A most straightforward quasi-derivation comes from Boltzmann's own H-Theorem which states: Where H(f) is defined as: The Boltzmann equations which can easily be derived from Louiville's theorem which states: Applying the Louiville's theorem here, for phase space distributions gives us the compact form: Extending, this to make it Lorentz invariant gives us (Clemmow and Wilson, 1957 Coming back to the H-theorem, we see that the only physically useful function which satisfies the Boltzmann equations is a probability distribution.Note, that there might be other functions which also satisfy the above criteria but are not relevant in this context.Thus, this function must be an invariant.However, engaging in a simple Jacobian of transformations from: Where the primed co-ordinates denote the relativistic scenario, we see that the integral is no longer of the same form. Thus, as pointed out, because of the additional factor, H(f) does not retain its form and needs suitable modifications, although the moot idea remains same.It's important to note though that is still invariant due to our above made assumption.
There is an important caveat to make here: Namely the Loschmidt paradox wherein Boltzmann supposedly obtains irreversibility using Louiville's theorem based on Newton's equations which are time symmetric.However, we will not digress and it will suffice to say here that this was resolved by assuming that initial states have low entropy which evolve to higher entropy over time.

Modified Helmholtz theorem
Campisi and Kobe (2010) demonstrated that for a Hamiltonian of a system given by (Campisi and Kobe, 2010): Here is the kinetic energy and p is the momentum.The particle is considered to be moving in a U-shaped potential as illustrated subsequently.Also note that V is some externally controllable parameter so that .Figure 1 is due to Campisi and Kobe (2010).
For a fixed V, the particle's energy E is a constant of motion.For simplicity, we define the zero energy in such a way that the minimum potential is 0, regardless of the value of V.
Once E and V are specified, the orbit of the particle in phase space is fully determined.Now applying Helmholtz theorem (Campisi and Kobe, 2010): A function S(E,V) satisfying the following equations exist and is given by: Where the equations to be satisfied are: The entropy can be written more compactly as: Where which is called the reduced action (Campisi and Kobe, 2010).It is basically the area in phase space enclosed by the orbit of energy E and parameter V, Where, What is remarkable about the above result is that it suggests that there exists a consistent one dimensional mechanical counterpart of entropy given by the logarithm of the phase space volume enclosed by the curve of constant energy To generalize the model to more degrees of freedom, original non-relativistic scenario, the below equation due to Clemmow and Wilson contain the additional term which although not apparent is simply due to the first order Taylor-Expansion of the Jacobian of transformation from .However, what is important is that this term will have a non-trivial effect on the resulting H-Theorem.

Figure 1 .
Figure 1.Point particle in the U-shaped potential =mV 2 x 2 /2.(a) Shape of the potential for a V=V1.(b) Shape of the potential for a V=V2.(c) Phase space orbit corresponding to the potential at energy E1.(d) Phase space orbit corresponding to the potential at energy E2.The two quantities E,V, uniquely determine one "state," that is, one closed orbit in phase space.