## Statistical mechanics: fundamentals and modern applicationsThis book examines the latest developments in statistical mechanics, a branch of physical chemistry and physics that uses statistical techniques to predict the properties and the behavior of chemical and physical systems. It begins with a review of some of the essential concepts and then discusses recent developments in both equilibrium and nonequilibrium statistical mechanics. It covers liquids, phase transitions, renormalization theory, Monte Carlo techniques, and chaos. |

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Page 343

the curves are composed of vanishingly small line segments that meet at sharp

angles. Since the

Similar remarks can be made for

If gaskets or sieves are formed by having holes systematically punched in them, a

...

**Fractal**curves, on the other hand, are continuous but nondifferentiable, becausethe curves are composed of vanishingly small line segments that meet at sharp

angles. Since the

**fractal**curves are not differentiable, they have an infinite length.Similar remarks can be made for

**fractal**surfaces and three-dimensional objects.If gaskets or sieves are formed by having holes systematically punched in them, a

**fractal**object such as the Sierpinski gasket (see Mandelbrot, 1982, p. 142; Figure...

Page 344

where D is called the

defined for a self-similar object. Solving for D, D = -lnN/lnr (13.11) An example of

a

curve is formed from a straight-line segment (called the initiator) by taking r = |.

The center one-third is removed and replaced by two continuous line segments,

each one-third ...

where D is called the

**fractal**dimension. Actually, what we shall refer to as the**fractal**dimension is more properly called the similarity dimension, since it isdefined for a self-similar object. Solving for D, D = -lnN/lnr (13.11) An example of

a

**fractal**curve is provided by the Koch curve shown in Figure 13.16. The Kochcurve is formed from a straight-line segment (called the initiator) by taking r = |.

The center one-third is removed and replaced by two continuous line segments,

each one-third ...

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For each application of the generator, r = I andiV = 3, which gives a

dimension of D — In 3/ In 2 = 1.58. 13.4.3

126 Starting from an ordered state at a single site with a value 1 , cellular

automaton 126 generates Figure 13.4. This figure illustrates both self-

organization and self-similarity. The figure is characterized by the basic unit

consisting of a large triangle surrounded by three smaller congruent triangles.

For purposes of determining the ...

For each application of the generator, r = I andiV = 3, which gives a

**fractal**dimension of D — In 3/ In 2 = 1.58. 13.4.3

**Fractal**Nature of Cellular Automaton126 Starting from an ordered state at a single site with a value 1 , cellular

automaton 126 generates Figure 13.4. This figure illustrates both self-

organization and self-similarity. The figure is characterized by the basic unit

consisting of a large triangle surrounded by three smaller congruent triangles.

For purposes of determining the ...

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### Contents

Classical Statistical Mechanics | 3 |

Quantum Statistical Mechanics | 45 |

Phase Transitions and Critical Phenomena | 83 |

Copyright | |

14 other sections not shown

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attractor automata average behavior bifurcation Boltzmann Boltzmann equation Brownian motion Brownian particle BZ reaction calculate called canonical ensemble cell cellular automaton chaos Chapter coefficient collisions configuration consider coordinates correlation functions coupling constants critical exponents critical point defined depends derivation dimension discussed distribution function dynamics eigenvalues entropy equilibrium evaluated example ferromagnetic fixed point fluctuations fluid FORTRAN FORTRAN program fractal free energy frequency Gaussian Hamiltonian initial integral interactions Ising model iterations Kramers Langevin equation lattice limit cycle linear magnetization Markovian matrix method microcanonical ensemble microstates molecular nonequilibrium nonlinear number of particles obtained one-dimensional order parameter oscillations partition function Pathria phase space phase transition polymer potential power spectrum probability density problem protein quantities quantum mechanics RANDOM NUMBER random walk scaling Section shown in Figure simulation solution spin glass statistical mechanics steps stochastic temperature theorem thermodynamic limit total number values vector velocity zero