Hidden subquantum processes as the true cause of the canonical distribution and the entropy

Transcription

1 Hidden subquantum processes as the true cause of the canonical distribution and the entropy V.A. Skrebnev Institute of Physics, Kazan Federal University, 18 Kremlyovskaya str., Kazan , Russia Abstract We show that the existing derivations of canonical distribution obscure the true nature of the physical processes in macrosystems; they are formal mathematical exercises based on assumptions which contradict physical reality. We also show that canonical distribution cannot be correctly derived within currently existing quantum mechanical theory. We offer a completely new derivation, free from unfounded assumptions and illuminating the true nature of canonical distribution. Canonical distribution is treated as a result of internal processes in macrosystems which are not described by the existing quantum formalism. It is these processes that are interrelated with entropy, the crucial characteristic of multi-particle systems. The study of these processes will further our knowledge of the laws of the microworld and macroworld. Key words: canonical distribution, alternative derivation. hidden subquantum processes, Boltzmann states. entropy. 1

2 Introduction As we know, along with many problems successfully solved by scientists, there are problems in science whose solutions under scrutiny turn out to be faulty. The longer such a problem finds no correct solution, the more painful it is to break long-standing faulty ideas. One such problem, indubitably, is the derivation of one of the most important physical formulas canonical distribution. Accepting an incorrect derivation of a correct formula strongly harms science and education, because it prevents the true understanding of the processes which the formula describes. Now the sad part: on September 5, 1906 the great physicist Ludwig Boltzmann committed suicide, hanging himself on a curtain cord. The standard opinion is that he was not understood by his contemporaries. That is not entirely true. The contemporaries understood that in his theories Boltzmann deviated from the requirements of mechanics. Boltzmann saw that too, but could not find a way out. Boltzmann derived his famous canonical distribution of ideal gas particles on energy levels as the most probable distribution when the number of particles and gas energy remain unchanged. However, according to Cauchy theorem, mechanics equations describing gas particles behavior under specified initial conditions can have only one solution. Since initial conditions always exist, by definition, then there ought not to be a probability of particles distribution. It would be required that initial conditions lead definitively to the most probable distribution. However, ideal gases arrive at canonical distribution under a variety of initial conditions. For example, when a ball is hit, initial conditions are created at the moment of the hit which do not guarantee the most probable distribution, but then the ball becomes round again. 2

3 Thus, Boltzmann s genius hypothesis regarding the most probable distribution has no justification within mechanics. However, this situation makes Boltzmann s hypothesis all the more meaningful. In spite of contradicting Cauchy theorem (and by extension, the strict laws of mechanics), Boltzmann s formula of canonical distribution is one of the fundamental formulas of science. Boltzmann could not provide a strict derivation of canonical distribution on the basis of classical mechanics not because he did not find it, but because it is impossible. It is equally impossible to arrive at canonical distribution while remaining physically correct and remaining within the existing quantum mechanics formalism. We offer a completely new derivation of canonical distribution. Canonical distribution is treated as a result of internal processes in macrosystems which are not described by the existing quantum formalism. We will call these processes subquantum processes. To understand this paper, a reader does not need to be an expert in a specialized area of physics. It is quite enough to know just the fundamentals of statistical physics and quantum mechanics. In turn, an attentive reading of this paper will help the reader better understand those fundamentals. Criticism of the traditional derivations of canonical distribution Textbooks on statistical physics use two models for deriving canonical distribution (see, for example, [1]). In one of the models the system is considered to be a part of a huge system. The environment of the system is often called the thermostat. But since the boundaries of the thermostat are unknown, as it is unknown what is beyond them, the total system is called the Universe, placed in quotation marks Universe ([1]). 3

4 The system s interaction with its environment is assumed to be extremely small, i.e. the system is practically isolated. At the same time it is assumed that the system has some energy value as a result of its interaction with the environment. It is also assumed that the system states are its eigenstates, which is possible only when the system s interaction with its environment is negligible. In accordance with this, the system s energy E is assumed to be one of the eigenvalues of its Hamiltonian. However in compliance with quantum mechanics the function of the system state, when interaction with the environment is negligible, can be represented as: (1) where are eigenfunctions of system Hamiltonian,. (2) In accordance with (1), quantum mechanical average system energy, i.e. its total energy, equals (3) Normalization requirement gives us the following: (4) If the number of the energy levels is more than two, equations (3) and (4) have a great number of solutions for. This means that the same value of energy corresponds to the great number of different system states. There are no reasons to ignore all states with the given energy except eigenstates. However, the paradox is that nobody tries to explain or is even interested in what happened to all the states with the given energy except the eigenstates. In the considered model it is supposed that the energy of "Universe" lies in the narrow interval Also, only eigenstates of the Universe are considered as the Universe states. All states of the Universe are assumed to be 4

5 equiprobable (however it is not explained what exactly provides the existence of the interval of energy and what processes cause the emergence of equiprobable eigenstates of "Universe" in this interval). Under this assumptions the probability of the system s eigenstates with the energy is found to correspond to canonical distribution [1]: =. (5) Equal probability of all states of the Universe corresponds to applying microcanonical ensemble to the description of the Universe. As we know, microcanonical ensemble describes systems in equilibrium. An inquisitive critically thinking textbook reader should notice that a living breathing observer who is near the system under consideration is a significant part of the system s environment. This means that the Universe is in equilibrium only if the observer has been dead for quite a while. In another model the Universe is assumed to consist of an enormous number of systems identical to the system under consideration [1]. The energy of these systems is assumed to be the eigenvalues of their Hamiltonian. Calculations are made for the most probable distribution of the Universe s total energy among its component systems. It turns out that the most probable distribution corresponds to the canonical distribution (5) in the system under consideration. Both models of the canonical distribution derivation arrive at the desired formula. However, neither the Universe, nor "Universe" are in equilibrium, nor do they consist of a great number of systems identical to the system under consideration, nor only the eigenstates of the system are possible. Thus, instead of reasonable physical models readers are given mathematical schemes which happen to lead to desirable results. A correct end result can be received by means of most 5

6 different, including delirious, assumptions (see collections «Physicists joke»). But it won't make the delirium true. Misconceptions in modern science can be more stunning than misconceptions of the past. It is wrong to consider that scientists of the past centuries were sillier than us. Scientific delusions of the past very often didn't contradict common sense. For instance, to be convinced that the Earth is flat, it is enough to live in an open country, and it is much more natural to assume that the small Sun which ascends in the east and sets in the west revolves round the huge Earth, rather the reverse. Today we have convinced ourselves and tell the students that it is possible to consider the Universe being in equilibrium or consisting of huge number of the systems identical to the studied one. Thus, we have enough bases to conclude, that physics has not found a satisfactory model allowing to obtain the canonical distribution as a consequence of interaction of the system with its environment. It forces us to reflect that the possibility of using the canonical distribution can be connected with internal processes in macrosystems. Canonical distribution as a result of hidden subquantum processes in macroscopic systems Here we offer a new derivation of canonical distribution that is free from amazing assumptions obviously contradicting the physical reality. The proposed derivation is based on the Boltzmann s method of the most probable distribution [2] and takes into account the internal processes which exist in physical systems and result in canonical distribution. 6

7 Remember that according to quantum mechanics the value in the equation (3) determines the probability of system energy being equal to. In accordance with (2) we have:. (6) We see that quantum mechanics does not allow the system to pass into a state with a set of different from the initial set. However, experience shows that in a macrosystem left to its own devices after an impact inducing certain initial conditions, the probability of the system s having energy after some time (the relaxation time) becomes described by canonical distribution, i.e. it does change. The speed of arriving at the canonical distribution does not depend on the properties of the surface of the macrosystem, nor on the structure of its environment. Thus, the influence of environment does not explain the transition of the probabilities of the system being in a state with energy to canonical distribution. This means that there must be internal processes which determine the transition of the initial distribution of probabilities to the canonical distribution. Hence, the canonical distribution may be derived as a result of internal processes of the macrosystem. The solution of Schrödinger equation is called the wave function. This function determines the distribution of probability of the values of the system s physical characteristics (e.g. the particle coordinates or energy). Quantum mechanics is a fundamental theory which allows to describe a vast number of physical phenomena. However, as a truly fundamental theory, it cannot explain and describe itself. Accordingly, quantum mechanics says nothing about internal processes within the physical systems which provide for the probability distribution. Therefore, we shall call them hidden subquantum processes. It is obvious that if these hidden processes did not exist, probability distribution would 7

8 not exist either. Thus, to understand quantum mechanics it is important to account for hidden subquantum processes in physical systems. The possibility of finding a particle in a certain point of space corresponds to one of the instantaneous states caused by the hidden subquantum processes. The wave function, which is a solution of the Schrödinger equation, allows to find the probability of those states. Since we are talking about probability of certain states, there must be transitions between those instantaneous states. Because transitions between instantaneous states exist, the states themselves also must exist. Hidden subquantum processes in physical systems must be very fast (recently [3,4] have shown that the lower boundary of the speed of the Einstein s spooky action at a distance is 10 4 light speeds). It is clear that a wave function may be used to describe the states of physical systems precisely because the hidden subquantum processes are extremely fast. This means that to describe the hidden subquantum processes we need a time scale whose graduation is many orders smaller than the usual. It follows from the above that the wave function describes the averaged state of the quantum system over times that are considerably longer than the times taken by the hidden subquantum processes. This is why the instantaneous system states and the mechanism of moving between those states ends up beyond the quantum mechanical description. We will use the term Schrödinger states for the averaged states of the quantum system which are described by the wave function. According to quantum mechanics the value in the equation (3) determines the probability of system energy being equal to. However, the value can not be considered the probability of eigenstates (see textbooks on quantum mechanics). Truly, according to (1) the quantum mechanical average value of any observable is 8

9 =, (7). If the value determined the probability of eigenstates, then we would have had = (8) Article [5] looks at the ensemble of system states with the same energy E. These states are considered as equiprobable. The entire set of equiprobable states is called the Generalized Quantum Microcanoncal Ensemble (GQME). Since in quantum mechanics the probability of the energy is determined by the value, one might have supposed that in a macroscopic system the value of averaged by GQME would correspond to canonical distribution. In article [5] all quantum superpositions of form (1) satisfying conditions (3) and (4) are considered to be equally probable. However, the averaging by this manifold performed in [5] has yielded the values of in a significant departure from the canonical distribution. The author of work [5] does not give a physical substantiation for the possibility of averaging by GQME. Obviously, we can speak about the probability of various states of a physical system only in the case when there are transitions between these states. However within quantum mechanics it is impossible to explain transitions between various states of GQME. It is obvious that some physical state of the system must correspond to every different. We can to treat as a probability of the system being in a state with energy and to use a formula (3) at such interpretation because during very small periods (on a usual scale) the system visits this state many times (see probability theory). It means that the system transitions with extremely high 9

10 frequency between the states with different. Consequently, every state with energy appears in the system for a very short time. We will use the term - states for such instantaneous states with energy. Importantly, -states should not be confused with the system s eigenstates which are described by wave functions the eigenfunctions of the system s Hamiltonian. -states which correspond to the same energy value may differ in other parameters. Transitions between -states, just as Einstein s spooky actions, are not described by quantum mechanics, but it does not mean that they do not exist in the physical reality. We call this type of hidden subquantum processes ghost actions. Because transitions between -states exist, the states themselves must exist in physical reality. The values of are proportionate to the average time of the system being in the states. To derive canonical distribution, we will use the method of the most probable distribution [1,2], but instead of the Universe consisting of an enormous number of macrosystems identical to the system under consideration we will consider a great number of events which take place in the single macroscopic system under consideration. Each event is a visit of the system to one of the instantaneous states with energy. Let N be the number of the system s cumulative visits of its instantaneous energy states over time t and let be the number of visits of -states, corresponding to the energy, over this time. Obviously, (9) Let s introduce the value. (10) 10

11 Numerous configurations determined by various sets of numbers of visits correspond to the value Each configuration may be realized in P ways corresponding to the number of permutations of the visits:. (11) In order to speak about probability of energy value which corresponds to -states it is necessary that the number of visits of these states during small periods (on a usual scale) was rather great. Then for valueν we can use Stirling approximation: (12) Using formula (12), we find the maximum of the function P (11) under conditions (9) and (10) and arrive at the most probable value of the numbers of visits of the -states:. (13) The probability of the system being in the -states equals the ratio of the number of the visits of those states to the total number of visits N: = =. (14) Thus, we have arrived at the canonical distribution. This shows that Boltzmann s genius hypotheses of the most probable distribution remains current and continues to stimulate fundamental research in physics. The fact that we obtained the correct formula of canonical distribution (14), using (12), once again tells us about extremely high frequency of hidden subquantum processes. We will use the term Boltzmann states for the averaged states of the quantum system which are described by canonical distribution (14). As we have just shown, 11

12 Boltzmann states correspond to the most probable distribution of the instantaneous energy states. Canonical distribution describes systems which have achieved equilibrium [1,6]. Arriving at an equilibrium is an irreversible process and thus cannot be described by Schrödinger equation, which is reversible in time. Here we need to note that there is no foundation for a belief that irreversibility is connected to the so-called dynamical chaos, since the mechanism of dynamical chaos is absent in quantum mechanics due to the strictly linear time evolution of the Schrödinger equation. As we know, even the best theory has applicability limits, beyond which it becomes inexact or even wrong. Both classical and quantum mechanics have resulted from the observation of systems with small number of objects. If the number of objects (e.g. particles) in a system is small and calculations are possible, the mechanics show amazing accuracy. One might assume that in systems with macroscopically great number of particles quantum mechanics would also be absolutely exact. However, this assumption contradicts the irreversibility of evolution of the macrosystems and the second law of thermodynamics. It is natural to consider that in macrosystems there exist the processes breaking the unitary Schrödinger evolution. We have shown above that when scientists attempt to derive canonical distribution without doubting the absolute exactness of quantum mechanics, it leads to physically absurd assumptions. God plays dice not exactly the way prescribed by Schrödinger. Schrödinger states of the system are called the pure states, while Boltzmann states are called the mixed states (see, e.g. [6]). To describe the mixed state we must use the density matrix, in which the non-diagonal components are assumed to equal zero in the energy representation, and diagonal components are described by (14). The appearance of Boltzmann states is an irrefutable experimental fact. The 12

13 irreversible process of the initial state of the macrosystem turning into a state with the most probable distribution (13) of the values and canonical distribution (14) is only possible if there exist irreversible transitions between -states which provide for this turning while the total energy E of the system is preserved. Let s call those transitions relaxation ghost actions or relaxation transitions (rtransitions). We view the r-transitions as manifestations of the internal properties of the macrosystems, which are not reflected in the existing quantum mechanics formalism. This paper points out that scientists should take into account those processes in the macroscopic systems which are not described by the unitary Schrödinger evolution. We have shown that the fundamental formula of canonical distribution can be obtained through the method of most probable distribution which accounts for such processes as relaxation ghost action, and does not require any physically unjustified assumptions. Нidden subquantum processes and entropy According to (14) the total energy of macrosystem Using equations (12) and (13) it is easy to show that for maximum P (15) (16) where S entropy of system (compare to the formula (61) of Boltzmann s paper [2]). In the textbook [6] entropy is defined as a logarithm of the number of eigenstates of a macrosystem defined from a condition (17) 13

14 When this condition is valid then (18) (in [6] it is designated as ). It is supposed that probability of the system to have energy has extremely sharp maximum at. It is obvious that the change of energy can be caused only by interaction of system with an environment. In the textbook [6] only eigenstates of system are assigned to each value of energy. Mechanisms of transitions between these states, and also frequencies of these transitions (which have to provide the possibility of using of some number as probability of a certain state), in [6] aren't discussed. Entropy is one of the fundamental concepts of science. Formula (16) connects entropy to the maximum number of system state realizations as a consequence of subquantum processes. We consider that the mixed states is the consequence of the macrosystem s tendency towards the maximum freedom in realizing its state with a given total energy, i.e. towards maximum entropy, expressed by (16). Conclusion Thus, one of the major formulas of physics, lying in the basis of statistical physics, currently is derived with the help of artificial assumptions which do not have any relation to physical reality. Instead of reasonable physical models, mathematical schemes are used to manipulate the achievement of desirable results. We have got used to these artificial schemes for the lack of others, but our habit can't make them true. Recognition of the fact of existence of the subquantum processes hidden today from our direct observation allowed us to obtain a derivation of canonical 14

15 distribution, free from the assumptions contradicting physical reality. We also offered a new interpretation of entropy as consequence of subquantum processes. We are confident that the irreversible processes occurring in the macrosystems are connected to r-transitions. In macrosystems, the canonical distribution nicely describes the results of experiments even with small measurement time. The decrease of the number of particles in a system can lead to the discrepancy between the measured values of the system parameters and their values calculated using the canonical distribution. This fact may allow to estimate the probability of r-transitions. The subquantum processes defining behavior of physical systems are one of manifestations of the general properties of a matter in the Universe. The study of these processes will further our knowledge of the laws of the microworld and macroworld. References [1] R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics (A Wiley- Interscience Publication, New York, 1975). [2] L. Boltzmann, Wiener Berichte 76, 373 (1877). [3] D. Salart, A. Baas, C. Branciard, N. Gisin, and H. Zbinden, Nature 454, 861 (2008). [4] J. Yin, Y. Cao, H.-L.Yong, et al, Phys. Rev. Lett. 110, (2013). [5] Boris V. Fine, Phys. Rev. E. 80, (2009). [6] L.D. Landau and E.M. Lifshitz, Statistical Physics, 3rd Edition Part 1 (Butterworth-Heinemann, Oxford, 1996). 15