Bogoliubov

The first tenn in the high-temperature expansion, is essentially the mean value of the perturbation averaged over the reference system. It provides a strict upper bound for the free energy called the Gibbs-Bogoliubov inequality. It follows from the observation that exp(-v)l-v which implies that ln(exp(-v)) hi(l -x) - (x). Hence... 

Ernst M FI 1998 Bogoliubov-Choh-Uhlenbeck theory cradle of modern kinetic theory Progress in Statistical Physics ed W Sung et al (Singapore World Scientific)... 

N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields, Inter-science, New York, 1959. 

Once this junction between the theory and the observed facts had been established, the subsequent work of fitting a number of known nonlinear phenomena into the framework of the theory of Poincar6 proceeded with an extraordinary rapidity in this initial stage (1929-1937) the work was done almost exclusively in the USSR. The western countries learned about this progress shortly before the beginning of the war, when two fundamental treatises on this subject, one by Andronov and Chaikin,4 and the other by Krylov and Bogoliubov,5 became available. After this, the work proceeded on an international scale. 

N. Krylov and N. Bogoliubov, Introduction to Nonlinear Mechanics, Kiev, 1937. 

The use of this theory in studies of nonlinear oscillations was suggested in 1929 (by Andronov). At a later date (1937) Krylov and Bogoliubov (K.B.) simplified somewhat the method of attack by a device resembling Lagrange s method of the variation of parameters, and in this form the method became useful for solving practical problems. Most of these early applications were to autonomous systems (mainly the self-excited oscillations), but later the method was extended to... 

In order to obviate this difficulty, Krylov and Bogoliubov suggested the transformation... 

In the Krylov-Bogoliubov version, this steady state solution is assumed in the form... 

In 1958 N. N. Bogoliubov and Y. A. Mitropolsky (B.M.) published a treatise entitled Asymptotic Methods in the Theory of Nonlinear Oscillations,18 which presents a considerable generalization of the early K.B. theory. Since a detailed account of this work is beyond the scope of this book, we give only a few of its salient points. 

Instead of seeking a simple periodic solution of the type of Eq. (6-89), Bogoliubov and Mitropolsky seek a solution of the form... 

Block relaxation, 61 Bogoliubov, N., 322,361 Boltzmann distribution, 471 Boltzmann equation Burnett method of solution, 25 Chapman-Enskog method of solution, 24... 

Kolmogorov, A. N., 114,139,159 Konigs thorem applied to Bernoulli method, 81 Koopman, B., 307 Roster, G.F., 727,768 Kraft theorem, 201 Kronig-Penney problem, 726 antiferromagnetic, 747 Krylov-Bogoliubov method, 359 Krylov method, 73 Krylov, N., 322 Kuhn, W. H., 289,292,304 Kuratowski s theorem, 257... 

One consequence of the positivity of a is that A A < (AU)0. If we repeat the same reasoning for the backwards transformation, in (2.9), we obtain A A > (AU)V These inequalities, known as the Gibbs-Bogoliubov bounds on free energy, hold not only for Gaussian distributions, but for any arbitrary probability distribution function. To derive these bounds, we consider two spatial probability distribution functions, F and G, on a space defined by N particles. First, we show that... 

The Gibbs-Bogoliubov inequalities set bounds on A A of (AU)0 and (AU) which are easier a priori to estimate. These bounds are of considerable conceptual interest, but are rarely sufficiently tight to be helpful in practice. Equation (2.17) helps to explain why this is so. For distributions that are nearly Gaussian, the bounds are tight only if a is small enough. 

These bounds are the nonequilibrium equivalents of the Gibbs-Bogoliubov bounds discussed in Chap. 2. Having the free energy now bounded from above and below already demonstrates the power of using both forward and backward transformations. Moreover, as was shown by Crooks [18, 19], the distribution of work values from forward and backward paths satisfies a relation that is central to histogram methods in free energy calculations... 

Thermofield dynamics Generalized bogoliubov transormations and applications to Casimir effect... 

Concepts in quantum field approach have been usually implemented as a matter of fundamental ingredients a quantum formalism is strongly founded on the basis of algebraic representation (vector space) theory. This suggests that a T / 0 field theory needs a real-time operator structure. Such a theory was presented by Takahashi and Umezawa 30 years ago and they labelled it Thermofield Dynamics (TFD) (Y. Takahashi et.al., 1975). As a consequence of the real-time requirement, a doubling is defined in the original Hilbert space of the system, such that the temperature is introduced by a Bogoliubov transformation. 

It should be noticed that a((3) and a((3) satisfy the same algebraic relation as those given in Eq. (3), and also that a(/3) 0(/ )) = a(j3) 0(/ )) = 0. Then the thermal state 0(/3)) is a vacuum for a((3) and a(/3) (otherwise, 0,0) is the vacuum for the operators a and a). As a result, the thermal vacuum average of a non-thermal operator is equivalent to the Gibbs canonical average in statistical physics. As a consequence, the thermal problem can be treated by a Bogoliubov transformation, such that the thermal state describes a condensate with the mathematical characteristics of a pure state. 

Abstract. Within the context of the Thermofield Dynamics, we introduce generalized Bogoliubov transformations which accounts simultaneously for spatial com-pactification and thermal effects. As a specific application of such a formalism, we consider the Casimir effect for Maxwell and Dirac fields at finite temperature. Particularly, we determine the temperature at which the Casimir pressure for a massless fermionic field in a cubic box changes its nature from attractive to repulsive. This critical temperature is approximately 100 MeV when the edge of the cube is of the order of the confining length ( 1 fm) for baryons. 

Keywords Thermofield dynamics, Bogoliubov transformation, Compactification,... 

In this talk, we consider the TFD approach for free fields aiming to extend the Bogoliubov transformation to account also for spatial compactification effects. The main application of our general discussion is the Casimir effect for cartesian confining geometries at finite temperature. 

The main goal of this talk is to show that the Bogoliubov transformation of TFD can be generalized to account for spatial compactification and thermal effects simultaneously. These ideas are then applied to the Casimir effect in various cases. 

These results demonstrate explicitly the usefulness of the Bogoliubov transformation to treat confined fields in the context of TFD. From the above considerations, a question emerges naturally what should be the appropriate generalization of the Bogoliubov transformation to account for simultaneously space compactification and thermal effects ... 

We have shown that generalizations of the TFD Bogoliubov transformation allow a calculation, in a very direct way, of the Casimir effect at finite temperature for cartesian confining geometries. This approach is applied to both bosonic and fermionic fields, making very clear the... 

It is important to stress that use of the generalised Bogoliubov transformatin provides an elegant physical interpretation of the Casimir effect as a consequence of the condensation in the vacuum of the fermion or the boson field. The method can be extended to other geometries such as spherical or cylindrical. 

Defining the time- and temperature-dependent annihilation and creation operators through the Bogoliubov transformation... 

Using TFD we are able to find the thermal expectation values of operators. In general, through the Bogoliubov transformation from ai (t), aa(t) to al(P,t),aa(P,t), we find the formula... 

Bogoliubov Laboratory of Theoretical Physics, JINR Dubna, 141980 Dubna, Russia... 

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