SELF

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S.B. Karavashkin and O.N. Karavashkina

3. Analysing the concept of conservative dynamic system

With account of these primary conclusions concerning the direction in which we have to improve the meaning of entropy, let us study the conception of conservative dynamical system.

It is known that under the term ‘conservative system’ we mean such selected volume or system of material bodies that have no energy exchange with their environment. We will not go into experimental details and will not point imperfect isolation, we will consider the issue on the whole. Although, on the other hand, as we could see it in the previous item, it is basically impossible to ensure a perfect isolation, especially in isothermal processes. In particular, “an isothermal process occurs at constant temperature (T = const). It follows from the Clapeyron – Mendeleev equation that for this process the Boyle – Marriott law is true:

To execute the isothermal process practically, we have to provide the thermal contact between the gas and massive body” [10, p. 203].

Let us think: is the process practically unrealisable because of imperfect isolation either because of imperfect studied body? Basically, “Inflow or outflow of the thermal energy can be caused by different physical phenomena. In applications, the following physical mechanisms of heat inflow are important.

1. Thermal conduction is the phenomenon of exchange of the average thermal energy between the parts of medium being in the immediate contact that occurs on the account of mechanical interactions and collisions in the heat motion of molecules, atoms, electrons and other particles composing the medium. Heat transfer caused by thermal conduction essentially relates to the macroscopically uneven distribution of temperature in the volume of bodies.

2. Heat radiation and absorption of radiation is the phenomenon caused by changes of possible states of particles (molecules, atoms, electrons etc.) of which the medium consists.

3. Heat emission caused by electric dissipative processes, in particular Joule heat released inside the body in presence of electric current.

4. Sometimes we can, with an additional stipulation, attribute to the external supply of heat dq^(e) some parts of increment of the internal energy dU and of the work of internal forces dA⁽ⁱ⁾, transferring these terms to the right part of heat supply equation. For example, the change of internal energy that takes place because of chemical transformations or phase transitions related to the heat emission or heat absorption can be substituted by the external heat flows. So we can consider only the change of exterior energy on account of change of temperature, mechanical parameters and, possibly, other changeable properties of medium” [9, p. 217].

In this list the fourth item is of our interest, it clearly differs from all previous. We see, this item admits existing the interior resources of thermodynamical system on whose account the state of this system can be changed independently of external sources, – and this is not inscribeable into the common conception of conservative system limited by the heat transfer processes. Just so Sedov suggested to exclude these interior resources of the system from the list of those interior, mere formally transferring the related terms in the heat transfer equation. But, of course, such formal mathematical operation may not change the phenomenology of process, it is able only to help solving the mathematical equations. In modelling, such sources anyway will remain interior, so it has to be reflected in the modelling and in determining the thermodynamical parameters. Without this, we will come to a discrepancy and will distort our physical understanding. We can easily show it, making Sedov’s transfer and analysing its consequences.

Suppose that we have transferred the interior sources of heat and wrote them as those exterior. First of all, with it we have to admit that we changed the interior energy dU by the value of these sources. But this is not all. According to the conventional idea that the conservative system does not contain interior sources, “if the exterior parameters remain unchanged (the work of external forces is zero), the energetic levels of the system remain unchanged, too. In this case the energy supplied to the system from the outside is spent to change the distribution of probabilities” [11, p. 392]. But the main is that “if then we isolate the system and leave it to its own, in some time the system inevitably will have to come to the state of static equilibrium. Actually, we said that the state of complex system does not depend on its initial state. So if the observation time is quite large, the main part of observation time the system is in the state of static equilibrium, irrespectively of, in which state it was initially. In some relaxation time the system that initially was disbalanced, a hardly probable state passes to a more probable, balanced state” [11, p. 392].

Thus, if the heat sources were exterior, we can isolate them in such or other way, and only having done so, we come to the stipulations on which the definition of entropy was based. But it is impossible to isolate the interior sources from the system, and far from always we can wait until their interior energy is exhausted. At the same time just these sources given as the stipulation of equilibrium of natural phenomena allow us breaking the equilibrium of thermodynamical system. To show it, consider a simple scheme. Select in some free space located far from heat sources and gravitating bodies some volume of intergalactic gas and filter it from the dust. The density of such gas is known to be one or two molecules per cubic millimetre and kinetic temperature is only about few Kelvin degrees, so we have all reasons to think this volume ideal. Having isolated this volume, we may surely state that in some time in this volume the thermodynamical equilibrium will be achieved, in full accordance with one direction of heat transfer. Having made sure that the equilibrium within the volume was achieved, attach to it just such volume and make sure that in some time in this aggregate volume the thermal equilibrium will take place. Let us go on attaching new volumes to a total size of parsec and larger. At a definite stage we will yield the process described by Shklovsky in his work [12], and the evolution of this process – in the second chapter of our work [13].

“Suppose, we have some cloud of the radius R whose density and radius R are constant. The condition at which the cloud will compress under affection of its own gravitation is the negative sign of the total energy of cloud. This last consists of the negative gravitational energy W_g of interaction of all particles forming the cloud and of the positive thermal energy of these particles W_T . The negative sign of total energy means, the gravity forces tending to compress the cloud exceed the forces of gas pressure tending to scatter this cloud in all the surrounding space. Then we have

where A = 8,310⁷ erg/molK , is the molecular mass, and is the average density of cloud. At the same time the gravity energy is

We see that W_T at the constant density of cloud and temperature T grows with R as R³, whilst W_g R⁵, i.e. it grows with R much faster. Consequently, at the given and T there exists such R₁ that at R > R₁ the cloud will inevitably compress under affection of its own gravity. When the mass M has been given, we can determine R₁ as

In this case (i.e., if the mass and temperature of the cloud have been given), if the size of cloud R < R₁ , it will compress.

It is easy to make sure that ‘usual’ clouds of interstellar gas with M M and R 1 parsec will not compress under their own gravity, but gas-dust complexes with M 10³ - 10⁴ M , T 50^o and radius about tens parsecs – will … Consider the case when the cloud mass is equal to the mass of Sun, and its temperature is 10 K. Then it follows from (8) that such cloud will compress if its radius is less than 0,02 parsec. Hence, the density of such cloud will be 210^-18  g/cm³, and the gas concentration in it is 10⁶ cm- ³ - quite considerable value. But if the cloud mass was 10 Solar masses, we can check that the average concentration of the gas particles at which the cloud starts compression will be well less, 10⁴ ?? ^-3… we really see the clouds with such concentration of gas” [1, p. 56- 57].

It follows from this description that if we enlarge our isolated volume with an ideal gas to the size at which (8) is true, we will come to the situation when the cloud is spontaneously compressed and at the same time heated and radiates the energy into its environment. With it, first, this process will be the same spontaneous as the heat transfer; secondly, it will be caused exceptionally by the internal forces of non-chemical nature. Chemical, thermonuclear etc. processes will begin not at the stage of cloud compression but due to the formation of a definite structure of the protostar. And the cloud will radiate from the very beginning of its compression. If now we, in accordance with the technique to find the entropy of this process, take as the temperature the initial temperature of gas, i.e. about several Kelvin degrees, and account that the system radiates the energy, we will yield

In other words, the entropy of isolated thermodynamical system containing an interior source is negative. And we cannot state that this source arises exceptionally when Shklovsky condition was true. The source shows itself at this condition, but the between-molecules interaction takes place also in small isolated volumes. Simply this interaction is non-critical. At the same time, just the complex between-molecules interaction causes the fact that, compressing this ideal gas, we can achieve its liquid phase. This will mean not the violation of ideality, in the standard meaning of absence of between-molecules interaction, but that the forces and interior sources that we disregarded when considering an ideal gas reveal themselves now. Namely the disregard of this fact causes the ruling opinion: “On the contrary, after the closed-loop system came to its equilibrium, the spontaneous overrun of the system from this state would be improbable” [4, p. 211]. While to all studies related to the system’s coming to the equilibrium there precede some actions that break the initial thermodynamic equilibrium of the system.

This conclusion says also that, considering an isolated thermodynamical system in its full meaning, we may not confine ourselves to the account of heat transfer but have to account also the interior sources that carry the system in balance. And vice versa, disregarding interior sources in case of local isolated thermodynamical systems, we may not generalise these results onto the common conception of isolated thermodynamical system. This common conception means not only one direction of thermal conduction from hotter bodies to those colder but also the presence of interior interactions between the molecules of medium that at definite conditions are able to localise the energy in some region of space, and the balance of these two determines the evolution of thermodynamic processes in the universe.