# Entropic Measure of Time, and Gas Expansion in Vacuum

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## Abstract

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## 1. Introduction

## 2. Model

_{0}and r, respectively (see Figure 1). The observer has no clock (timepiece). However, the purpose of the observer is to introduce a measure of time based on the properties of the system of particles around him. Using the above-mentioned properties of the measure of time and following this paper’s purpose, the observer wants to use the entropy method. The entropy that can be introduced for arbitrary, including essentially non-equilibrium, systems [3] would be naturally chosen. For the problem under consideration, the entropy change equals the entropy production (no entropy flows through the system’s boundaries). Since only spatial changes occur in the system, the observer divides the system’s current size r into G similar cells having the size Δ (Figure 1). This size remains unchanged and, additionally, can contain up to N particles. It is obvious that, at different system-observation instants, G varies and equals 2r/Δ. The size of the cell selected by the observer depends on the degree of detail in which the observer prefers to describe the system. Based on the information which is known by the observer (or, as we should rather say, a total lack of information about the system, including details of its initial state), all possible distributions of particles by cells should be considered equally probable. This probability is inversely proportional to the number of such distributions Ω. This number by which N identical particles may be arranged by G cells with an arbitrary number of particles per cell is well known [18]:

## 3. Entropic Measure of Time

_{0}, r, N, G, and Δ. The outside observer will calculate the system’s specific entropy in exactly the same manner as in Equation (5). However, originally having a clock, the observer can introduce the velocity of motion υ = dr/dt. Therefore, for the considered one-dimensional motion, G = 2(r

_{0}+ υt)/Δ. As a result, according to Equation (6),

- Let $t\to 0$, then$$\mathsf{\tau}\propto {\mathsf{\tau}}_{1}+{\mathsf{\xi}}_{1}t/{t}_{1}$$
- Let $t\to \infty $, then$$\mathsf{\tau}\propto \mathrm{ln}\left(\text{\hspace{0.05em}}t/{t}_{1}\text{\hspace{0.17em}}\right)$$

We have not a direct intuition of the equality of two intervals of time. The persons who believe they possess this intuition are dupes of an illusion. When I say, from noon to one the same time passes as from two to three, what meaning has this affirmation? The least reflection shows that by itself it has none at all. It will only have that which I choose to give it, by a definition which will certainly possess a certain degree of arbitrariness.

…there is not one way of measuring time more true than another; that which is generally adopted is only more convenient. Of two watches, we have no right to say that the one goes true, the other wrong; we can only say that it is advantageous to conform to the indications of the first.

- Previously, the one-dimensional case was considered. It can be easily extended to a three-dimensional one. The gas expands radially with the spherical symmetry. Initially, it is contained in a sphere with the radius r
_{0}and then occupies ever larger spheres of the radius r. As before, let us write the number of cells and the number of particles as G and N, respectively. As in the case above, we will assume the volume of cells is constant and designate it as Δ^{3}. Given the symmetry of the problem, the shape of the cells can be selected as spherical layers of some thickness centered on the system’s point of symmetry. Obviously, the thickness of these layers is to decrease in inverse proportion to r^{2}. For such a formulation, G = 4π(r/Δ)^{3}/3. It is evident that the expression for the time measured by the inside observer using the variables G and N will remain the same as Equation (6). However, the relation of the inside observer’s time and the reference time has the form:$$\mathsf{\tau}=\left(1+{\left({\mathsf{\alpha}}_{3}+t/{t}_{3}\right)}^{3}\right)\mathrm{ln}\left(1+{\left({\mathsf{\alpha}}_{3}+t/{t}_{3}\right)}^{3}\right)-3{\left({\mathsf{\alpha}}_{3}+t/{t}_{3}\right)}^{3}\mathrm{ln}\left({\mathsf{\alpha}}_{3}+t/{t}_{3}\right)$$$$\mathsf{\tau}\propto {\mathsf{\tau}}_{3}+{\mathsf{\xi}}_{3}t/{t}_{3},t\to 0$$$$\mathsf{\tau}\propto 3\mathrm{ln}\left(\text{\hspace{0.05em}}t/{t}_{3}\right),t\to \infty $$Thus, as is seen from the given formulae, the three-dimensional case has no fundamental differences from the linear one. - Previously, we have considered a strongly non-equilibrium case of gas expansion in vacuum. This case cannot be investigated using the methods of classical thermodynamics. It can be shown, nevertheless, that for a number of adjacent problems, results similar to the ones above can be obtained thermodynamically using a number of restrictions. We will consider some initial thermodynamic equilibrium state of an ideal gas. The gas adiabatically expands from it in vacuum and reaches another thermodynamic equilibrium state while changing the volume V. Obviously, this process is irreversible and the gas does no work. According to the adiabatic nature of the process and the first law of thermodynamics, the temperature of an ideal gas is to remain unchanged during expansion. Let us replace a real irreversible process with a hypothetic isothermal equilibrium process having identical initial and final states. The changes of thermodynamic entropy S
_{t}for the two processes are the same and it is easy to show [1,18] that the change of entropy in the case at hand is equal to$$d{S}_{t}=\mathsf{\nu}R\text{\hspace{0.05em}\hspace{0.17em}}dV/V$$

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**The simplest model of identical particles spreading out in vacuum considered herein. The initial state and one of the following states are shown. All symbols in the figure are explained in the text hereof.

**Figure 2.**Dependence of the entropic time τ introduced by the observer inside the system on the time chosen by the outside observer as the uniform time t. The dashed lines show limit approximations Equation (11) at small and large times. t

_{1}= 0.005, α

_{1}= 1000.

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Martyushev, L.M.; Shaiapin, E.V.
Entropic Measure of Time, and Gas Expansion in Vacuum. *Entropy* **2016**, *18*, 233.
https://doi.org/10.3390/e18060233

**AMA Style**

Martyushev LM, Shaiapin EV.
Entropic Measure of Time, and Gas Expansion in Vacuum. *Entropy*. 2016; 18(6):233.
https://doi.org/10.3390/e18060233

**Chicago/Turabian Style**

Martyushev, Leonid M., and Evgenii V. Shaiapin.
2016. "Entropic Measure of Time, and Gas Expansion in Vacuum" *Entropy* 18, no. 6: 233.
https://doi.org/10.3390/e18060233