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# Boltzmann's Law's of Statistical Mechanics

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a)
i) Boltzmann's Entropy-Probability Relation: The entropy of a system is a function of the probability of the state of the system.

Where S is the Entropy of the system, k is Boltzmann's Constant and is the thermodynamic probability of the state of a system, or in other words, the total number of microstates corresponding to the given macrostate of the system.

Boltzmann Transport Equation: This equation describes the statistical distribution of one particle in a fluid. The Boltzmann equation describes the time of evolution of the distribution or the density function in one-particle phase space.

is the force field acting on the particles in the fluid, and is the mass of the particles. describes the density function in one-particle phase space, where x and p are position and momentum, respectively. The term on the right hand side is added to describe the effect of collisions between particles. If it is zero then the particles do not collide.
The Boltzmann equation is used to study how a fluid transports physical quantities such as heat and charge, and thus to derive transport properties such as electrical conductivity, Hall conductivity, viscosity, and thermal conductivity.

ii) Phase Cell: The phase space is an imaginary six-dimensional space in which the six coordinates of position and momentum, namely are marked along the six mutually perpendicular axes in space. The phase space is further divided into tiny six-dimensional cells whose sides are given as . Such cells are called phase cells.
The volume of each of these phase cells is given as

The thermodynamic state of a classical gas can be described by specifying the distribution of molecules of the gas among the various phase cells. The probability of occurrence of all possible distributions that are permitted by the thermodynamic state of the gas can be determined. The state of a system, when it is in thermodynamic equilibrium, corresponds to the most probable distribution of particles in the phase space.

b)
i) Boltzmann's Distribution Law: Boltzmann's Distribution Law states that if the energy associated with some state or condition of a system is then the frequency with which that state or condition occurs, or the probability of its occurrence, is proportional to

Where T is the absolute temperature of the system and k is Boltzmann's constant.

Boltzmann derived a relationship which states that the natural logarithm of the ratio of the number of particles in two different energy states is proportional to the negative of their energy separation.

Where Ni is the number of molecules at equilibrium temperature T, in a state i, which has energy Ei and degeneracy gi, N is the total number of molecules in the system and k is the Boltzmann constant. Because velocity and speed are related to energy, the above equation can be used to derive relationships between temperature and the speeds of molecules in a gas. The denominator in this equation is known as the canonical partition function.

ii) The temperature of the gas is required in order to work out the equilibrium distribution of molecular speeds.

Reference:
www.scienceworld.wolfram.com/physics/Maxwell-BoltzmannDistribution.htm
Statistical Physics- L.K.Pathria
www.wikipedia.com

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