Computational Statistical Mechanics by Wm.G. HOOVER (Eds.)

By Wm.G. HOOVER (Eds.)

Computational Statistical Mechanics describes using speedy desktops to simulate the equilibrium and nonequilibrium houses of gases, drinks, and solids at, and clear of equilibrium. The underlying thought is built from uncomplicated rules and illustrated through using it to the best attainable examples.

Thermodynamics, in keeping with the proper gasoline thermometer, is said to Gibb's statistical mechanics by using Nosé-Hoover warmth reservoirs. those reservoirs use crucial suggestions to manage temperature. a similar procedure is carried via to the simulation and research of nonequilibrium mass, momentum, and effort flows. this type of unified procedure makes attainable constant mechanical definitions of temperature, tension, and warmth flux which result in a microscopic demonstration of the second one legislation of Thermodynamics at once from mechanics. The intimate connection linking Lyapunov-unstable microscopic motions to macroscopic dissipative flows via multifractal phase-space buildings is illustrated with many examples from the hot literature.

The booklet is well-suited for undergraduate classes in complicated thermodynamics, statistical mechanic and delivery thought, and graduate classes in physics and chemistry.

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Example text

Find the order of this "method" and show that it is unstable for the onedimensional harmonic oscillator. Then show that the alternative method, q + + « q + + q_ - q__ + (dt 2 /4)[5a+ + 2a 0 + 5a_], is stable for an oscillator. Compare the "computational efficiency" [accuracy at a fixed time with a fixed number of force evaluations] of this latter method to that of the fourth-order Runge-Kutta method. In the simplest cases the kinetic energy depends only on the momenta and the potential depends only on the coordinates.

For a Hamiltonian with the form H(p,q) = E(p2/2m) + O(q) the momenta {p} can be eliminated from these difference equations by calculating the second differences, { q+ - 2q + q_}. The result is the set of Stoermer equations: G { q+ - 2q + q_ = dt (F /m) = (dt/m)[p+ - p ] } . } along an exact continuous trajectory, for a Hamiltonian which differs from the specified Hamiltonian H by terms of order the time step dt. To illustrate this correspondence for the simplest interesting case, consider a one-dimensional harmonic oscillator with the ratio (K/m) chosen so that Stoermer solution is: q(t) = cos(cot); p(t) = -cosin(cot) + (dt/2)cos(cot); (K/m)(dt ) = 2- 2cos(codt); co = 1 - (dt/2) .

It is assumed that the pistons have no thermal energy, only gravitational energy. The thermodynamic equation of state for the pistons is simply E = mgh. The thermodynamic equation of state for the three-dimensional ideal gas follows from the definition of an ideal-gas thermometer: P 0 V 0 = NkT 0 = (2/3)E 0 , where the subscripts indicate initial-state conditions. The nonequilibrium process proceeds by converting the mechanical energy of the falling piston to thermal energy of the gas. The process is complicated.

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