Loschimidt's Paradox and Causality: Can we establish Pearlian expression for Boltzmann's H-theorem?

Boltzmann (Wikipedia)

Preamble

Probably the most important achievement for humans is the ability to produce scientific discoveries, that  helps us objectively understand how nature works and build artificial tools where no other species can.  Entropy is an elusive concept and one of the crown achievements of human race. We question here if causal inference and Loschmidt's paradox can be reconciled. 

Mimicking analogies are not physical

Before even try to understand what is a physical entropy, we should make sure that there is only one kind of physical entropy from thermodynamics, formulated by Gibbs-Boltzmann ($S_{G}$ and $S_{B}$).  Other entropies such as Shannon's information entropy are all analogies to physics, and mimicking concepts.

Why counting microstates are associated with time?

The following definition of entropy is due to Boltzmann but Gibbs' formulation tend to provide equivalence, technically different formulations aside, they are actually equivalent.

Definition 1: An entropy of a macroscopic material is associated with larger number of states its constituted elements take different states, $\Omega$. This is associated with $S_{B}$, Boltzmann's entropy.  

Now, as we know from basic thermodynamics classes that entropy change of a system can not decrease, so the time's arrow. 

Definition 2: Time's arrow is identified with change in entropy of material systems, that $\delta S \ge 0$.

We put aside the distinction between open and close systems and equilibrium and non-equilibrium dynamics, but concentrate on how come counting system's state's are associated with time's arrow? 

Loschimidt's Paradox: Irreversible occupancy on discrete states and causal inference

The core idea probably can be explained via discrete lattice and occupancy on them over chain of dynamics. 

Conjecture 1: Occupancy of $N$ items on $M$ discrete states, $M>N$, evolving with dynamical rules $\mathscr{D}$ necessarily increases $\Omega$, compare to the number of sampling if it were $M=N$. 

This conjecture might explain the entropy increase, but irreversibility of the dynamical rule $\mathscr{D}$ is required addressing Loschimidt's Paradox, i.e., how to generate irreversible evolution given time-reversal dynamics. Actually, do-calculus may provide a language to resolve this, by inducing interventional notation on Boltzmann's H-theorem with Pearlian view. The full definition of H-function is a bit more involved, but here we summarise it in condensed form with a do operator version of it.

Conjecture 2 (H-Theorem do-conjecture): Boltzmann's H-function provides a basis for entropy increase, it is associated with conditional probability of a system $\mathscr{S}$ being in state $X$ on ensemble $\mathscr{E}$. Hence, $P(X|\mathscr{E})$. Then, an irreversible evolution from time-reversal dynamics should use interventional notation, $P(X|do(\mathscr{E}))$. Then information on how time reversal dynamics leads to time's arrow encoded on, how dynamics provides an interventional ensembles, $do(\mathscr{E})$.

Conclusion

We provided some hints on why would counting states lead to time's arrow, an irreversible dynamics.  In the light of the development of mathematical language for causal inference in statistics, the concepts are converging. Along with understanding Loschmidt's Paradox via do-calculus, it can establish an asymmetric notation. Loschmidt's question is long standing problem in physics and philosophy with great practical implications in different physical sciences.

Further reading

• Loschimid's Paradox
• H-Theorem
• do-Calculus revisited, J. Pearl (2012) pdf
• Causal Inference : Looper Repository for collection of resources.
• H-theorem do-conjecture, M. Süzen,  arxiv:2310.01458 (2023) 

Please cite as follows:

 @misc{suezen23lpc, 

     title = {Loschimidt's Paradox and Causality: Can we establish Pearlian expression for Bolztmann's H-theorem?}, 

     howpublished = {\url{https://science-memo.blogspot.com/2023/02/loschimidts-do-calculus.html}}, 

     author = {Mehmet Süzen},

     year = {2023}

}  

@article{suzen23htd,

    title={H-theorem do-conjecture},

    author={Mehmet Süzen},

    preprint={arXiv:2310.01458},

    url = {https://arxiv.org/abs/2310.01458}

    year={2023}

}

Insights into Bekenstein entropy with an intuitive mathematical definitions: A look into Thermodynamics of Black-holes

Jacob Bekenstein
(Wikipedia)

Preamble

Thermodynamics of black holes has appeared as one of the most interesting areas of research in theoretical physics [Wald1994], specially after LIGO's massive success. The striking results of Jacob Bekenstein  [Bekenstein1973] in proposing a formulation of entropy for a black hole was on of the most striking turning point in building explanations for the thermodynamics of gravitational systems. Bekenstein entropy is defined to be so-called a phenomenological relationship and surprisingly easy to understand concept using basic dimensionality analysis. In this post, we will show how to understand the entropy of a black hole just using basic dimensionality analysis, fundamental physics constants and basic definition of entropy. 

Dimensions and scales

Dimensionality analysis appears in many different areas of physics and engineering, from fluid dynamics to relativity. The starting point is to understand the concept of dimensions. Every quantity we measure in real life has a dimension. It means a quantity $\mathscr{Q}$  we obtain from a measurement $\mathscr{M}$ has a numeric value $v$ and associated unit $u$. $\mathscr{Q}=\langle  v, u \rangle$ given $\mathscr{M}$. There are 3 distinct fundamental unit types length (L),  time (T) and mass (M).

Intuitive Bekenstein entropy (BE) for a black hole : Informal mathematical definition

Black holes are astronomical objects that are not directly observable due to their mass condensed in a small area. The primary object we will use is something called Planck length $L_{p}$ and it implies physically possible smallest patch of the space-time, this is associated with the state of the black holes on their horizon. We won't define the Planck length here in detail but with the knowledge of fundamental physics constants and dimensional analysis we mentioned, one can get a constant value for this length. 

Definition: Finite entropy $S_{f}$ of an object is associated with the number of states $\Omega$ a system can attain.

If we combine this definition for a black hole entropy : 

Definition Finite entropy of a black-hole $S_{f}^{BH}$ is  associated with the number of its states $\Omega$, number of elements on it's surface area of $A$. The elements are discretised with  small patches $a_{p}=L_{p}^{2}$. Then intuitively,  $\Omega$ yields to $A$ divided by $a_{p}$.

  

Bekenstein entropy is not thermodynamic entropy alone and family of Bekenstein entropies

The unit analysis tells us that $A$ has a dimension of length square.  We intentionally omit any equality in the above definition upon $S_{f}^{BH}$ because, in practice Bekenstein Entropy is not thermodynamic entropy alone. The formulation usually presented as BE in general uses equality for the above approach. However this is not strictly thermodynamical alone, that's why we specify definitions as finite entropy and only express the relationship as association. Similarly any other constants as it can yield to different Bekenstein entropies such as introduction of new constants would yield to family of Bekenstein entropies.

Why surface area defines states of a black-hole?

This is an amazing question and Bekenstein's main contribution is to associate this to number of states of a black-hole on event horizon, i.e., point of of no return layer whereby ordinary matter can't return. The justification is that all other properties of black hole defines this surface. Here is the intuitive definition of states of black-hole.

Definition A surface area $\mathscr{A}$ is formed by the set of physical properties forming an ensembles. such as charge density, angular momentum. These ensembles indirectly samples thermodynamics ensembles. 

Even though intuition is there, this question might still be an open question further.

Conclusion

We provided the primary idea that Bekenstein tried to convey in his 1973 paper intuitively. However,  we identify its thermodynamic limit is an open research area. Thermodynamic limit implies that taking infinite limit of both area and the discretised areas, even though it sounds that the values might converge to infinity, simultaneous limit would converge to a finite value for a physical matter. 

Primary Papers

• Bekenstein J.D.: Lettere al Nuovo Cimento, 4, 737, (1972)
• Bekenstein J.D.: Physical Review D, 7, 2333, (1973)  
• Bekenstein J.D.: Physical Review D, 9, 3292 (1974)
• Bekenstein J.D.: Physical Review D, 12, 3077 (1975)

Primary Book

• Wald, Quantum field theory in curved space times (1994) 

Please cite as follows:

 @misc{suezen23ibe, 

     title = {Insights into Bekenstein entropy with an intuitive mathematical definitions}, 

     howpublished = {\url{https://science-memo.blogspot.com/2023/02/bekenstein-entropy.html}, 

     author = {Mehmet Süzen},

     year = {2023}

  }

Postscript A: 

Information can’t be destroyed

Proposals of that information is destroyed out of thin air is a red flag for any physical theory: this includes theories on evaporating black holes. Bekenstein’s insight in this direction that surface area is associated with entropy. The black-holes’   information in this context is quite different than the Shannon’s entropy. An evaporating black-hole, the area approaching to zero is not the same as information going to zero, surface area is a function of  physical properties of the stellar object that bound  by conservation laws in their interaction with their surrounding. Hence, the information is preserved even if area goes  to zero.

Postscript B: 

What is Holographic principle? its origins from Bekenstein Entropy perspective

The word embedding applies in this context as well. Embedding implies some sort of  dimensionality projection. A projection to lower dimensional space, or on the other end,  to the higher dimensional space. Holography is no different. Imagine taking 2D snap shots of rotating 3D objects, generating this in reverse is the end effect of holographic  reconstruction. N-dimension to (N-1) projection. This is the bases of holographic principle: entropy of black-holes doesn’t appear as all states of its constituted matter,  as normally should have for ordinary matter, it manifest as N-1 projection on it’s surface. This kind of holographic entropy is first noted by Bekenstein; whereby he assigned the event-horizon area as a representation of the states of the black-hole volume. This projection to (N-1)-dimension is improved upon Bekenstein’s approach to generalised situations in explaining how universe might be  a hologram entirely by Gerard 't Hooft and Leonard Susskind. Holographic principle, probably one of the most important development in theoretical physics in recent times.