Ising-Conway lattice-games: Understanding increasing entropy

Preamble

The entropy is probably one of the most difficult physical concepts to grasp. Its inception roots in efficiency of engines and foundational connection to multi-particle classical mechanics to thermodynamics,  i.e., kinetic theory to thermo-statistics. However, computing entropy for a physical systems is a difficult task, as most of the real-physical systems lacks the explicit formulation. Apart from advanced simulation techniques that invokes thermodynamical expressions, pedagogically accessible and physically plausible system is lacking in the literature. Addressing this, we explore here, recently proposed Ising-Conway Games.

Figure: Evolution of Ising-Conway
Game  (arXiv:2310.01458)

Ising-Conway Lattice-Games (ICG)

Ising-Lenz model is probably one of the landmark models in physics, remarkably provides beyond its idealised case of magnetic domains,  now impacts even quantum computational research. However, computing entropy of Ising-Lenz models are still quite difficult. On the other hand, Conway introduce a game with simple rules generating complexity in various orders, via simple dynamical rules. By analogy to these two modelling approach,  we recently introduce game like physical system of spins or lattice sides on a finite space with constraints. This gives a physically plausible dynamics but simpler dynamical evolution to generate the trajectories. Because vanilla Ising-Models requires more complicated Monte Carlo techniques.  Here is the configuration and dynamics of Ising-Conway games,

1. $M$ sites as a fixed space.
2. $N$ occupied sites, or 1s.  
3. Configuration $C(M,N,t)=C(i)$ over time changes. But at $t=0$ all occupied sites live in at the corner.
4. Configuration can only change to neighbouring sites if they are empty. This is closely related to spin-flip dynamics of the Ising Model. 
5. No sites occupy the same lattice cell, Pauli exclusion
6. Should be contained within $M$ Cell.

An example evolution is shown on the Figure.

Defining ensemble Entropy on ICG

Now we are in position to define the entropy for ICGs, which easy to grasp conceptually and computationally.  $C(i, t) \in \{1,0\}$ defines the states of  the game. We build an ensemble at a given time $t$ by defining a region enclosed by 1s.  Then dimensionality of the ensemble  $ k(t) = argmax[\mathbb{I}(C(i))] - argmin [\mathbb{I}(C(i)) ]$. Here,  $\mathbb{I}$ returns index of $1$s on the lattice. This ensemble closely track maximum entropy of the system at a given time. 

Conclusions

A new game-like system that helps us to understand entropy increase that has a plausible physical characteristics that one can easily simulate.

Further reading

• H-theorem do-conjecture, M.Süzen, arXiv:2310.01458
• Effective ergodicity in single-spin-flip dynamics, Mehmet Süzen. Phys. Rev. E 90, 03214 url
• do_ensemble module provides such simulation via simulate_single_spin_flip_game  from the repo h-do-conjecture 

Please cite as 

 @misc{suezen23iclg, 

     title = {Ising-Conway lattice-games: Understanding increasing entropy}, 

     howpublished = {\url{https://memosisland.blogspot.com/2023/10/ising-conway-games-entropy-increase.html}}, 

     author = {Mehmet Süzen},

     year = {2023}

}