By R.K. Pathria (Auth.)

ISBN-10: 0080541712

ISBN-13: 9780080541716

ISBN-10: 0750624698

ISBN-13: 9780750624695

'This is a wonderful e-book from which to benefit the tools and result of statistical mechanics.' Nature 'A good written graduate-level textual content for scientists and engineers... hugely steered for graduate-level libraries.' selection. This hugely winning textual content, which first seemed within the yr 1972 and has endured to be well known ever considering that, has now been introduced up to date by means of incorporating the striking developmentsRead more...

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**Sample text**

3iV, (1) dqi where H(qi, pt) is the Hamiltonian of the system. Now, as time passes, the set of coordinates (^/, p/), which also defines the microstate of the system, undergoes a continual change. Correspondingly, our representative point in the phase space carves out a trajectory whose direction, at any time t, is determined by the velocity vector V = (qi, pi), which in turn is given by the equations of motion (1). /)]. In particular, if the total energy of the system is known to have a precise value, say E, the corresponding trajectory will be restricted to the "hypersurface" H{qi,pi) =E (2) of the phase space; on the other hand, if the total energy may lie anywhere in the range (£ — | A, £ + ^ A), the corresponding trajectory will be restricted to the "hypershell" defined by these limits.

Whence E [(3iV/2)-l]! 17, 17a), we obtain the desired correspondence, viz. 9. Quite generally, if the system under study has ^ (^ degrees of freedom, the desired conversion factor is a;o = / i ' " . (3) In the case of a single particle, ^ 1 = 3 ; accordingly, the number of microstates available would asymptotically be equal to the volume of the allowed region of the phase space divided by h^. Let E{P) denote the number of microstates available to a free particle confined to volume V of the physical space, its momentum p being less than or equal to a specified value P.

7) ds ¥ The next case we shall consider here is that of a one-dimensional simple harmonic oscillator. The classical expression for the Hamiltonian of this system is Hiq,p) = ^kq^ + ^ p \ (8) where k is the spring constant and m the mass of the oscillating particle. The space coordinate q and the momentum coordinate p of the system are given by q =A cos (cot + <^), p = mq = —mcoA sin (cot + 0), (9) A being the amplitude and co the (angular) frequency of vibration: CO = y/(k/m). (10) The energy of the oscillator is a constant of the motion, and is given by E = \mco^A^.

### Statistical Mechanics by R.K. Pathria (Auth.)

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