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Einstein’s quantum oscillator
in an equilibrium state:
Thermo-Field Dynamics or
Statistical Mechanics?
Prof. A.Soukhanov,
Russia, OGOL@OLDI. RU
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Setting of the problem
As it is known the Einstein’s initial theory of fluctuations covers only the
domain of thermal fluctuations at values of temperature T
determined by an inequality
kB
– is the Boltzmann’s constant;
τ
– is a specific time interval for the system.
For other values of temperatures T and intervals τ an essential role
play or quantum fluctuations (the subject of Quantum Mechanics)
either they with thermal ones simultaneously. But now there is no
successful theory joined both types fluctuations.
Meanwhile there is a great needs in the such theory. Open quantum
systems are of interest. Among them are systems with a few number
of freedom degrees by a low temperature and systems with unstable
ground state. There are some reasons to suppose that quantum and
thermal fluctuations are correlated among themselves and their
values are comparable.
where
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The aims of our studies are
modification of normal thermodynamics and
extension of Einstein’s fluctuations theory to the case
when thermal and quantum uncontrollable influences
are enjoying equal right.
As a tool of our studies we use Schroedinger’s
uncertainties relations (SHUR), which allow us to
consider not only non-commutation but as well
correlation of conjugated observables (characteristics)
of system. They reflect themselves geometrical
properties of the corresponding space and therefore their
using assigns an universal meaning to description of
Nature.
As a model of the open quantum system we use a quantum
oscillator in the thermostat (thermal bath).
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Some important results
- an inconsequence of an approach in the frames of the
Quantum Statistical Mechanics to description of quantum
oscillator in the thermostat is established
- an advantages of an approach based on Thermo-Field
Dynamics are shown
- a confirmation of existing of thermal noise in the pure
thermal state is found
- a generalized definition of entropy is suggested
- the canonical distribution is modified by means of
entering of generalized temperature
- a program of generalization of Einstein’s fluctuations
theory and creating of the Quantum
StatisticalThermodynamics is planned
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The main Hypothesis
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Taking as a model of an open system the oscillator we will assume that   
when ω – is a frequency of classical oscillator.
We will represent thermostat as infinite set of sequences of N identical bound
quantum oscillators with frequencies  in interval 0    ,where N .
The Hypothesis: a quantum oscillator might be in thermal equilibrium state with
the thermostat at any temperature including T=0. In that case the energy of
system will fluctuate even at T=0 due to a resonant binding between vacuum
oscillators and the system.
As it is known the average value of energy of quantum oscillator in thermostat is
at high temperatures
at low temperatures
For description unification
let us suppose in this Formula at T0
where TW is the Wigner temperature (1932) contained in the Wigner function for
quantum oscillator at T = 0 .
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Some special features of SHUR
We shall deal with oscillator in the thermal equilibrium
state at T=0 to examine some special features of SHUR
in this case.
For observables coordinate q and momentum p it has a
form
, where
is the sum of the
correlator square
and
commutator square
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Some calculations
Calculating separately the left and right hand sides of SHUR at the
ground state one will find
Thus in this case at the condition
SHUR takes the
form of strong equality, named saturated SHUR.
Let us assume that the saturation of SCUR for momentumcoordinate is a property of thermal equilibrium state at any
temperature even if σqp ≠ 0.
Note that there is a peculiar dependence between
observables p and q even in this state as their dispersions are
proportional to the Wigner temperature.
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A hypothesis verification by means
of Quantum Statistical Mechanics
Suggesting that perhaps
at any temperature SHUR for oscillator in
thermostat should be also a saturated
Let us check this hypothesis
We start with using of the usual Quantum Statistical
Mechanics in a mixed state. In this theory:
the temperature of a system in thermal equilibrium
state is rigorously equal to the thermostat
temperature;
coordinate and momentum of the system are
considered as independent observables.
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In this case the coordinate and momentum dispersions are increasing together
with temperature
But the right-hand part of SHUR does not change under heating, since
and
Nevertheless the dispersions q and p are proportional to each other
but σqp is not dependent on the temperature. We focus attention on this
contradiction!
Thus SHUR at T>0 stops to be saturated, and the Quantum Statistical
Mechanics becomes useless for solving of our problem, as expected.
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A hypothesis verification by means
of Quantum Mechanics
Now let us try to use the Quantum Mechanics for our aim. It must be
admitted that saturating of SHUR it is possible not only by the
condition
.
There exist a collection of pure states with
for which SHUR is saturated. They are called correlated coherent
states (CCS) or squeezed states.
Among these are all complex wave functions of Gaussian type.
where α is any function of time independent on coordinate.
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Obtaining a saturated SHUR
At this state we obtain
The independent calculation of right-hand side of SHUR gives us
;
As one can see we obtain just the same expression in the both sides
The SHUR is saturated as expected !
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A verification by means of
Thermo-Field Dynamics
Let us find among this states a wave function corresponding to the thermal
equilibrium state. With another words we have to determine a concrete α
This problem might be successful solved by methods of Thermo-Field-Dynamics
(TFD). This is a variant of Quantum Field Theory at finite temperature in real
time created by Umezawa.
It assumes thermostat as well as vacuum are systems with infinite number of
freedom degrees.
The main idea of TFD:
Under the influence of thermostat appear two independent possibilities of energy
absorption by a system:
-either an excitation of new quanta;
-either a filling of new vacancies.
Thus, putting of a system into the thermostat is equivalent to an effective doubling
of freedom degrees number. It results in cutting of a peculiar degeneration of
state. Therefore we transfer from initial vacuum for particles |0> to a new
vacuum for quasi-particles |0>>, which is dependent from temperature.
It is possible to do it by means u-v – transformation of Bogoljubov for a system
with infinity number of freedom degrees.
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Obtaining of α
In the TFD framework it is possible to show
that a pure thermal state in thermal
equilibrium is fixed by value of α
Now the right-hand side of SHUR takes the
form
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The analysis of TFD results for
quantum oscillator (1)
A). From the saturated SHUR for observables p and q it follows
The quantity
has a meaning of an effective quantum of action dependent
on T. In particular at T>>TW we have
Then a statistical weight 
We think of as the transition

should be observed by experiment
and be an essential part of a new generalized theory like TFD at T0.
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The analysis of TFD results for
quantum oscillator (2)
B). The obtained value of
allows to say that
dependence of the wave function phase on temperature takes place. Namely
It means that we face with existence of "thermal noise" in a pure state. It remains to be
valid at T>>TW .
C).This fact is contrary to the definition of entropy suggested by von Neumann. According
it the entropy in the pure state must be equal to zero.
We propose to change this definition in order to account a contribute of phase dependent
on temperature. In our opinion the generalized definition of entropy might have the
following symmetric form:
The phase contribution is accounted indirectly for the second term in this sum.
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The analysis of TFD results for
quantum oscillator (3)
Then for a system at the pure state in the thermostat with wave
function
we obtain
A new essential feature of this quantity at T=0
(the Third Principle!)
where 0 ≡ e has a sense of the statistical weight at T=0. This is an
universal quantity – it has the same value for any oscillators.
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A program for generalizing of
Thermodynamics (1)
On the basis of these considerations we propose a program for
generalizing of Thermodynamics – transition to Quantum
Statistical Thermodynamics
In the traditional Quantum Statistical Mechanics one starts from the
solving of quantum dynamics problem without considering of
thermostat and then a system is placed into the thermostat.
We propose using an inverse consequence of actions:
to start from the entering of thermo-field vacuum having the
temperature Tgen and
then to put a system into it describing at the same time a system
behavior by classical means.
A model of thermostat at the temperature T:
an infinite set of independent sequences of N bounded quantum
oscillators having a frequency , where N is an infinite number and
0   .
Thereby a vacuum vector
is an infinite product of independent
vacuum vectors
specific to each oscillator sequence.
The system under study (if it can be approximated by quantum oscillator)
interacts by resonant means with a proper sequence of vacuum
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oscillators.
A program for generalizing of
Thermodynamics (2)
In all formulas of thermodynamics we propose to use the
Planck average energy of quantum oscillator
as an alternative to the average energy of classical
oscillator
Taking for unification of our description
we obtain
where Tgen is the generalized temperature for the first time
referred by Bloch (1932).
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Generalization of formulas
We think of as the generalization of thermodynamics may be carried out if we substitute for
generalized temperature Tgen instead T into the all formulas of usual thermodynamics:
the generalized distribution of Gibbs
the generalized free energy
the generalized entropy
Let us note that the calculation of entropy gives us the same value as calculation by mean the
formula for a pure state in the thermal equilibrium.
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Some results in the frames of this
program (1)
Using the generalized canonical distribution (for
mixed classical states) one can calculate the
correlator of coordinate and momentum
By means of TFD (for pure states) we have
obtained
Despite of Cgen =0
we see that
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Some results in the frames of this
program (2)
II. The fluctuations of energy and temperature have the form
From this formulas we can see that even at T 0 the energy and the
frequency of ground state have fluctuations
Thus the idea of the thermal state for the quantum oscillator gets an
adequate description in this approach even at T=0
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Some results in the frames of this
program (3)
III. Following to this way we suggest also the expression for generalized formula of
Carno
Let us note that here TW is a common characteristic both for heater and for condenser
because it is connected with the normal mode of vacuum oscillator which is equal to
the frequency of the oscillator playing a role of operating substance.
At T2 ;T1 >> TW we get the usual formula
In conclusion may be said that there is real a difference between Quantum
Statistical Mechanics and new Quantum Statistical Thermodynamics.
In our opinion using of generalized temperature open new possibilities for applications of
the fluctuations theory to many problems in a sufficiently wide temperature domain
and calculations of entropy for many objects.
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