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Effective action in the quantum generalization of Equilibrium Statistical Thermodynamics

A.D.Sukhanov,

O.N.Golubjeva

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References

• O. Golubjeva, A. Sukhanov. Some geometrical analogues between the describing of the states space in non-classical Physics and the events space in classical Physics. VI Intern. Conference BGL-08. Debrecen, Hungary, 2008 (in print)

• A. Sukhanov. Some consequences of quantum generalization of statistical thermodynamics . XIII Intern. Conference “Problems of quantum field theory”. Dubna, Russia, 2008 (in print)

• A. Sukhanov. A quantum generalization of equilibrium statistical thermodynamics: effective macroparameters. TMP, 154 (1), 2008

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• A. Sukhanov. Schroedinger uncertainties relation for a quantum oscillator in a thermostat. TMP, 148, 2 . 2006

• A. Sukhanov. Quantum oscillator in a thermostat as a model thermodynamics of open quantum systems. Physics of particles and nuclei. 36, 7a, 2005

• A. Sukhanov. Schroedinger uncertainties relation and physical features of correlated-coherent states. TMP, 132 (3), 2002

• A. Sukhanov. On the global interrelation between quantum dynamics and thermodynamics. XI Int. Conf. “Problems of quantum field theory”. Dubna, Russia, 1999

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Contents • Introduction • Operator of effective action• Correlate-coherent states and effective

action vector• Effective action as a macroparameter• Effective action for QO in quantum

thermostat• Effective action and thermodynamic Lows• On the Third Low of Thermodynamics• Conclusion• References

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Introduction • In the last years there appeared some

additional experimental and theoretical reasons showing that the minimal value of entropy in thermodynamic equilibrium is not equal to zero. Particularly, in the quantum generalization of equilibrium statistical thermodynamic (QEST) recently supposed by us the effective entropy minimal value is

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• But this fact is in contrary to the standard result of usual quantum equilibrium statistical mechanics (QESM)) according to it entropy must be equal to zero at the limit T0.

• Let me shortly remind line of reasoning that give such result in QEST. In our theory a quantum thermostat at an effective temperature T* is introduced. It characterizes quantum and thermal influence of environments simultaneously

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• In its frame the effective entropy for quantum oscillator in thermal equilibrium is expressed by a new macroparameter – the effective action J as

• For introduction of the quantity J in our previous paper we used some heuristic consideration from classical mechanics (particularly – notion of adiabatic invariants) allowing us in fact to postulate it in the form

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• where

is the energy of QO in thermal equilibrium (according to Planck). We see that at T 0 the effective action for QO runs up to its minimal value

and

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• We emphasize that the approach used in QEST is a macroscopic one. Besides it was only approved on a model QO. As a result there exists some dissatisfaction. We think it would be better to have more firm evidences in support of this step, i.e. of introduction of quantity J that has a deep physical meaning. Besides we would like to extend QEST ideas to another microobjects in thermal equilibrium.

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• That is why, we are trying to look at this problem on the other hand. Now we are starting to move from a microscopic theory to confirm that we are on the correct way.

• We proceed from the assumption that so far apart approaches as micro- and macro-theories, can give us agreed results.

• After this introduction let me pass to the matter.

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Operator of effective action

• If we know a wave function (q) that is usually for microscopic approach we have a possibility to directly count the quantity J determining the corresponding operator on the Hilbert space of states but not to postulate it! As some heuristic reason for this goal can serve an analysis of mutual dependent fluctuations of coordinate and momentum on the base of Schroedinger uncertainties relation (SUR).

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As is well known their dispersions are given by formulas

• = ; = where • and

are fluctuations operators of coordinate and momentum. Next we use the CBS-unequality in respect to states submanifold

• The subject of our special interest is the term in the right side of the expression.

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• The quantity

has a sense of an transition amplitude from the state to the state . At the same time it is the Schroedinger correlator and equals the mean value of an operator in an arbitrary state. Considering that the quantity R has non-zero value in any non-classical theory (like QM, ST,etc ) we can claim that the given operator is meaningful. Taking also into account its dimension we can thereafter call it “effective action” operator

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• Of course, we remember operators

and are non-commutative. Besides their product is non- Hermitian one.

• Taking the operator in the form

we can select here the two Hermitian terms• and

where is the unique operator

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• It is easily to see that the operator reminds the expression of standard fluctuations correlator of coordinate and momentum in the classical probabilities theory. It reflects a contribution of stochastic environment influence into the transition amplitude Rpq because it includes the both fluctuations operators. That is why we will call it operator of “ external effect”. We remark that it was earlier discussed by Schwinger.

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• At the same time the operator does not have any analogies out of quantum dynamics. It reflects the universal feature of microobject "to feel" stochastic influence of “cold” vacuum and to react under it. It is proportional to unique operator. It means that any state transforms to the same state under its using. Therefore it can be called "own action” operator . Its mean value does not depend on the state and always has the constant value for any microobject

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• We are of opinion, all this information is a sufficient reason to claim that we have deal with an universal concept. We suggest to give it the special name - the own action. Its very value is already eloquent because it is connected with such well known quantity as the fundamental Planck constant .

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• However as some remark we would like to tell you some words of terminology because usually one calls “elementary quantum of action”. But in J0 there is the meaningful coefficient ½. And we know that there is not a half of quantum. Hence, in our opinion, J0 and

might have some different physical sense and different names.

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The effective action as a macroparameter.

• It is obviously that the mean value of operator

coincides with the transition amplitude R pq and it is a complex quantity

with a phase .

Its modulus

we consider as macroparameter and call simply "effective action" .

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• In quasi-classical limit (at ) Rpq goes to the real quantity. In this limit operators and can be changed by c-numbers so becomes correlator

from the classical probabilities theory.

• The significance of quantity J is in the fact that it is of paramount important in the right side of SUR "coordinate-momentum"

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Correlate-coherent states and effective action vector

• In a number of important cases this expression takes a form of strict equality (the saturated SUR)

• The subject of our interest in this talk are states in which the saturated SUR are realized

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• If at the same time 0 we have deal with correlate-coherent states (CCS). Among such states one can select a family of thermal CCS. They describe microobjects in thermal equilibrium.

• Besides there exist states called the simplest coherent states (CS) for that the term is absent ( for example, the basic state of oscillator at T=0 or the initial state of wave packet).

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• We can pass from each this CS to CCS using (u,v)- Bogoliubov transformations with parameters depending either on temperature or on time. The given transformations generate Lie-group SU (1,1) which is local isomorphic to Lorenz-group in two-dimensional events space.

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• It means we can consider a set (J,) as two-dimensional time-like vector in some pseudo-Euclidian space and the quantity as a length of the vector or the invariant of corresponding group

It is easily to see that effective action "vector" (J,) is an analog of two-dimensional vector energy-momentum ( , p ) in relativistic mechanics

(For more convenience we put here с=1).

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• As well known the given expression determines a surface of hyperboloid on pseudo-Euclidian momentum space called a mass shell.

• Analogically we can assume that the expression for the vector (J,σ) also gives us a shell as some hyperboloid surface in the pseudo-Euclidian space. It seems to us that the presence of the invariant right side says of some stability in respect of such states inducing corresponding shell .

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• Taking into account that using the Bogoliubov transformations for QO we come to thermal CCS now we could connect this stability with some thermal equilibrium at a conventional effective temperature T*. That is why such a shell could be called thermal “equilibrium shell”.

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Effective action for QO in quantum thermostat

• Let us use the information obtain above for behavior investigation of a quantum oscillator in a thermostat. This model allows us to calculate macroparameter J making direct averaging of corresponding operators because earlier in our preceding paper we had obtained a wave function for QO in thermostat.

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It has the form

Here

Temperature T here fixes one of possible thermal states i.e. plays a role of parameter.

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Then one can calculate mean values of interest

and

It is self-understood that the equality

is ensured that is pleased..

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Thus we recognize the results precisely coinciding with the sequence predicted by QEST. We only remind that J was introduced there as a ratio between QO energy and its frequency. We are of opinion that this fact can serve as a good argument in support of our effective action definition .

So, the concept of effective action is universal one. We emphasis it is not connected with any concrete object. By the example of QO we have got confirmations that the given concept also has a microscopic base.

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Effective action and thermodynamic Lows• Now let us return to the thermodynamics

language. In our opinion with a fair degree of confidence we can extend the employment region of this quantity. We suppose that interconnection between effective action and effective temperature T* having place for QO in the form

still stand for any microobject in thermal equilibrium. But we emphasis here one can calculate J as mean value of operator . .

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Thus on the base of the proportionality starting from this moment we will write the Zeroth Low of thermodynamics, as it seems to us, in the more general form

• In so doing we translate the fluctuations idea to the effective action. Here J and ΔJ are the effective action and its fluctuation of an object in thermal equilibrium but Jeq. is the effective action of a quantum thermostat.

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• In other words we claim that in generalized thermal equilibrium mean effective actions of two contacting objects (of a microobject and a quantum thermostat) coincide.

• Using imaginary we repeat the well known aphorism “action is equal to contra-action” in its literal sense. We are of opinion that such a way of thermal equilibrium describing is more adequate to nature than operating not only with effective T* but even with customary (however rather abstract) notion as temperature T.

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• Going consecutively on this way we can rewrite all thermodynamics formulas of QEST through the quantity J including the canonical distribution function in the macroparameters space as well as the First and the Second Lows.

• Having the all necessary formulas one can also obtain the expression for the effective entropy

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We remark following:• -it resembles in appearance to the formula

written above (in the beginning of the talk). But there is a very essential difference by the sense of the quantity J that here is the mean value of the effective action operator;

• -now it has more wide sense for it can be associated with any microobjects but not only with QO;

• - according to it the minimal value of effective entropy S*min. for any object is k B as before

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On the Third Low of Thermodynamics

• The last fact demands some additional discussion that is quite appropriate here because we have the proper instrument for dispute in hands. It is useful to remind that there are statements in literature according to them a minimal value of some “quantum” entropy Squ. can be equal to non-zero but does not equal to k B .

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• An usual approach to calculating of minimal quantum entropy Squ min. bases on the formula

• Here and amplitudes squared of wave functions in basic state in q- and p- representations.

• But we are of opinion that it is unsuitable for this aim for under the sign of logarithm it has dimensional terms.

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• So we suggest to introduce new dimensionless variables

where is an arbitrary numeral constant. Then the new dimensionless amplitudes squared of wave functions are

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• After that we get the expression for minimal quantum entropy in the form

• If we substitute in it the wave function for quantum oscillator in the basic state and its Fourier-image we obtain the minimal quantum entropy

• This result means that the answer depends on the factor

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We can consider the three variants as it is shown in the table:

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• Thus the result is determined by a constant choosing and in the frame of purely quantum dynamics we can not get the single-value one. But this choose we can make single-value if we compare the two minimal values: of the effective

and of the quantum . It is easily to see that they coincide at giving us

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We are pleased getting one and the same result either from micro- and macro- theories. In our opinion, this fact is of great importance:

• On one hand we have made some contribution in the solving of the problem with the constant in the Nernst theorem;

• On other hand we have confirmed the significance of Boltzmann constant – it gives us the minimal value of entropy.

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Conclusion • As a conclusion we allow ourselves to make

some remarks going out of the frame of the given concrete results.

• We think that the idea of the operator effective action introduction is fruitful because it gives a chance to describe objects moving nonperiodically.

• The agreement between thermodynamic (in the frame of QEST) and quantum (in the frame of TFD) approaches says, in our opinion, that micro- and macrodescriptions on the right footing can be used for the solving of problems with quantum-thermal phenomena simultaneously.

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• Our proposed theory additionally elevates the constant kB status from usual dimensional factor to the level of the fundamental constant of macrodescription.

• At the same time we emphasis that the

quantity is also meaningful as the fundamental constant of microdescription.

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• From theoretical point of view, it would be very interesting to establish interrelation between the notions of spin projection and effective action as the physical characteristics connected with rotation either in the Euclidian or pseudo-Euclidian plane.

• It would be good to obtain some new experimental confirms of fundamental status of the two constants and k В as such as well as joining the both these quantities.

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References

• O. Golubjeva, A. Sukhanov. Some geometrical analogues between the describing of the states space in non-classical Physics and the events space in classical Physics. VI Intern. Conference BGL-08. Debrecen, Hungary, 2008 (in print)

• A. Sukhanov. Some consequences of quantum generalization of statistical thermodynamics . XIII Intern. Conference “Problems of quantum field theory”. Dubna, Russia, 2008 (in print)

• A. Sukhanov. A quantum generalization of equilibrium statistical thermodynamics: effective macroparameters. TMP, 154 (1), 2008

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• A. Sukhanov. Schroedinger uncertainties relation for a quantum oscillator in a thermostat. TMP, 148, 2 . 2006

• A. Sukhanov. Quantum oscillator in a thermostat as a model thermodynamics of open quantum systems. Physics of particles and nuclei. 36, 7a, 2005

• A. Sukhanov. Schroedinger uncertainties relation and physical features of correlate-coherent states. TMP, 132 (3), 2002

• A. Sukhanov. On the global interrelation between quantum dynamics and thermodynamics. XI Int. Conf. “Problems of quantum field theory”. Dubna, Russia, 1999