Violating the second law of thermodynamics in a dynamical system through equivalence closure via mutual information carriers of a 5-tuple measure space

Violating the second law of thermodynamics in a dynamical system through equivalence closure via mutual information carriers of a 5-tuple measure space Deep Bhattacharjee Project Director of AATWRI – R&D Directorate of Electro-gravitational Simulation & Propulsion Laboratory, Bhubeneshwar, Odissa, India Database Administrator at PhyxBuzz, Kolkata, West Bengal, India Email: itsdeep@live.com, hi@phyxbuzz.com Abstract – Time and space average of an ergodic systems following the 5-tuple relations ( , ~, , Σ, ) through the initial increment from + to + + indicates the entropy to be reserved in the deterministic yet dynamical and conservative systems to hold for the set = ∑∞ keeping as the entropy ∃( ∞ = ⋯ = ) > obeying the Poincar ́ recurrence theorem throughout the constant attractor . This in turn states the facts of the equivalence closure as the property of the induced systems to resemblance an entropy conserving scenario.

with a more probably conservative limit that the system generates over a finite period of time. It is nevertheless crucial to determine the nature of such a function where the system is both dynamical and conservative, thus being ergodic. This is the prospects of this paper. Just as, any equation can be plotted as a graph, a special type of equation has been discussed here where there is the capacity of a central attractor around which the points of the equation evolves in accordance with the limits of the variables. This in turn imposes the system to act repeatedly taking the shape over and over without losing its track between the initial and the final states. Thus, the points which are monodromic around the attractor might revolve infinitely taking its initial pattern over and over again. Therefore, one might suggest that, the function itself acts over the space where the entropy is increasing with each iterations, but it has been shown in this paper that, such a case does not exist over the modified Archimedean spiral equations that have been considered in the function space. Therefore, the initial state when jumps to its next periodic state may give rise to many other, rather numerous periodic states without naturally losing any kind of entropy. That being said, the entropy has been conserved throughout the process. Both the case of joint and conditional entropy has been considered while making this paper and it has been deliberately proved that, its only through the equivalence closure, that the function stops its entropy from getting increased over time thus violating the 2 nd law of thermodynamics. Saying that, it needs to be considered that, entropy is also the 'quantized' information carriers where the mutual information from one state to another is being conveyed by in the modified equations of + + + + ⋯ + . This mutual information that has been transmitted by the function over the period of time makes the entropy conserved through the repeated iterations. Set theoretic approach of analysis has been provided in this paper to depict the states of the function through a series of five diagrams where the base points ( , ) has been made as the initial pole from where the spiral is generated by a generator that continues up to infinity, provided the values are bounded over a certain interval. It has also been demonstrated that the system loses its track of dynamicity and behaves as a circle that expands and contracts depending on the determinants of the spinner and spreader.
Methodology -To measure the dynamical evolution of a system in a geometric space, think of the system as a particle and space as a state where a typical temporal evolution term acts on all points of the space through both time and space evolution operators that goes on the 5-tuple relations ( ,~, , Σ, ) where each item is defined as [1];  is the attractor of the space of the evolving system where every union takes place surrounding its boundary point defined as ∪ ∪ … in the set where the initial evolution parameter keeps track of all the positions from onwards in the space thereby conserving entropy over the whole set .
 ~ is the equivalence closure where ∐ /~ ∈ forms the locus of the periodicity that makes the boundary points evolves keeping the entropy conserved with the initial state as the final state.
 Σ is the -algebra on the set .
The pole in the equation = + + which acts as the equation = + at the base points ( , ) can be stated providing = 0, = can be expressed as; Provided we are getting three kernels to map the vector field in the singularity as such; And the function runs through 4 identities to satisfy as below in order; The part is invariant taking all the partial sums of ℓ(∑ ∞ ) while the latter part derives the telescoping series for a relation as is a measure preserving endomorphism of on the operator space as noted by the relations; ( , ) = , ∞ Which satisfies the limit of the series as given below taking all the four relations; Therefore, in the measure space, ( ,~, , Σ, ), provided ( ) is finite and nonzero, the sojourn time [11] can be defined as the time spent on the set having the Haar measuring function ( ) defined as, Where is the indicator function of set with two elements {0,1} ∃ ⟶ 1 is probable in the measuring space ( ,~, , Σ, ) which is the number of iterations over provided for the initial point ≅ ( , ) = 0 ∃ = − given as; While the ergodicity converges to = Φ ≅ ∞ upon the metric space in time expressed as; 1 The singularity being satisfied as ( , ) Φ the constant entropy is preserved without any singularity as; ( , ) Therefore, the relation between the spinor and the spreader can be defined as a matrix product; The determinant if can be computed of the matrix then this yields to the result; This satisfies the inclusion of the base points at the starting of the ergodicity through a deterministic curve of repeated iterations as the corresponding base point equations to be developed in the way by inclusion of into ( , ) as;  For the equation + , is the mutual information carrier between and . Given, and random values, carries the mutual dependency between them which is the amount of information that can be obtained from one random variables by imposing authority on the other variables, its not necessary that the variables stick to two, it can be upto infinity but as we state earlier the variables are amplifiers where one variable amplifies the other like in the equation + + , amplifies and while coordinates the information between them, which is linked to the 'entropy' where information of the random variable is quantized. Now, in the Kullback-Leibler [16] divergence notation this can be defined over the function of the entropic space (or curve) ℓ( ) × ℓ ∑ ∞ of a pair of the points, ( , ) × ∑ ( , ) entropy takes place and gets conserved by means of the mutual information carriers, that is, making a way out for the sets to close equivalently via equivalence closure theorem to guide the elaborate process of the evolving dynamical system over repeated iterations in a conserved means through a local attractor. More emphasis has been made with respect of the mean and the support set, where through the summation convention, the relation of equivalence closure has been properly diversified by the spinor and spreader matrix determinants in the crucial base points over repeated revolutions around it. This process can run infinitely with a more probabilistic amplitude of the correct divergences that loops around the base points like its previous thereby maintaining the property of the conservative (more precisely ergodic) systems in a 5-tuple measure space through the generators of the space.
Declaration: Author of this paper has no conflicting interests.
Funding -No funding has been issued for the purpose of this work.