Kinetic Theory and Statistical Mechanics, Physics tutorial

Introduction:

Define: The Statistical Mechanics gives the connection between the microscopic motion of individual atoms of matter and macroscopically observable properties like temperature, pressure, free energy, entropy, heat capacity, viscosity, chemical potential, spectra, reaction rates and so on.

The fundamental problem in statistical mechanics is to bridge the gap between the huge thermodynamic properties of matter and a classical, or quantum mechanical, explanation of the microscopic systems of which it is composed.

On one hand we encompass the thermodynamic properties of the macroscopic sample, like temperature, pressure, energy, volume, heat capacity, entropy and so forth. Thermodynamics explains a high class of phenomena we examine in the macroscopic systems. The main purpose of statistical mechanics is to account for this behavior in terms of the dynamical laws regulating the microscopic elements of macroscopic systems and probabilistic suppositions.

Statistical physics was founded around in the year 1870 by Boltzmann, Maxwell and Gibbs (by employing the classical physics). Whenever quantum mechanics appeared, the common ideas of statistical mechanics could be adopted having no major complexity. Statistical physics exerted to the macroscopic bodies is closely associated to thermodynamics that was as well developed throughout the 19th century.

Why do we need Statistical Mechanics?

1) Statistical Mechanics gives the microscopic base for thermodynamics, which, or else, is merely a phenomenological theory.

2) Microscopic basis lets computation of a broad variety of properties not dealt by in thermodynamics, like structural properties, by employing distribution functions and dynamical properties - spectra, rate constants and so on, by using the time correlation functions.

3) As statistical mechanical formulation of a problem starts by a detailed microscopic explanation, microscopic trajectories can, in principle and in practice, be produced providing a window to the microscopic world. This window frequently gives a means of connecting some macroscopic properties having specific modes of motion in the complex dance of the individual atoms that compose a system, and this, in turn, allows for interpretation of the experimental data and an elucidation of the methods of energy and mass transfer in the system.

Fundamental postulates of statistical mechanics:

1) Microstates (that is, individual quantum states) corresponding to the similar macroscopic state variables (E, V, p and so on) can't be differentiated.

2) Ensembles (Gibbs) set of all the systems having the same given macroscopic state variables comprises an ensemble. The observed outcome of a macroscopic measurement is the average over the equivalent ensemble.

3) Equal priory probabilities: Each and every reachable quantum state in a closed system is uniformly probable.

4) Equilibrium in the macroscopic system corresponds to the most probable macrostate consistent by using the given constraints.

Microscopic laws of motion:

Assume a system of 'N' classical particles. The particles confined to a specific area or region of space by a container of volume 'V'. The particles encompass a finite kinetic energy and are thus in constant motion, driven through the forces they use on one other (and any external forces that might be present). At a given instant in time 't', the Cartesian positions of the particles are r1(t), ..... rN(t) . The time evolution of the positions of the particles is then represented by Newton's second law of motion:

m1r ‾i = Fi (r1... rN)

Here, F1... FN is the forces on each of the 'N' particles due to all the other particles in the system. The notation rˆi = d2ri/dt2

The ensemble concept (heuristic definition):

For a common macroscopic system, the total number of particles N ~ 1023. Since a fundamentally infinite amount of precision is required in order to state the initial conditions (due to exponentially rapid growth of errors in this specification), the amount of information needed to identify a trajectory is essentially infinite. Even if we contented ourselves with quadrupole precision, though, the amount of memory required to hold just one phase space point would be around 128 bytes = 27 ~ 102 bytes for each number or 102 x 6 x 1023 ~ 1017 Gbytes.

Ensemble: Consider a large number of systems each explained by the similar set of microscopic forces and sharing some general macroscopic property (example: the similar total energy). Each and every system is supposed to evolve beneath the microscopic laws of motion from a different initial condition in such a way that the time evolution of each and every system will be different from all the others. Such a collection of systems is termed as an ensemble. The ensemble theory then states that macroscopic observables can be computed via performing averages over the systems in the ensemble. For most of the properties, like temperature and pressure that are time-independent, the fact that the systems are evolving in time will not influence their values, and we might carry out averages at a specific instant in time. Therefore, let 'A' represent a macroscopic property and suppose 'a' represent a microscopic function that is employed to calculate 'A'. An illustration of 'A' would be the temperature, and 'a' would be the kinetic energy (that is, a microscopic function of velocities). Then, 'A' is obtained through computing the value of 'a' in each and every system of the ensemble and functioning an average over all systems in the ensemble:

A = (1/N) λ=1ΣN

Here 'N' is the total number of members in the ensemble and 'aλ' is the value of a in the λth system.

Boltzmann's formula:

A very significant thermodynamic concept is that of entropy 'S'. Entropy is the function of state, similar to that of the internal energy. It computes the relative degree of order (as opposed to disorder) of the system when in this state. The understanding of the meaning of entropy therefore needs some appreciation of the way systems can be explained microscopically. This connection between the thermodynamics and statistical mechanics is preserved in the formula due to Boltzmann and Planck:

S = k ln Ω

Here 'Ω' is the number of microstates accessible to the system (that is, the meaning of that phrase to be illustrated).

We will at first try to describe what a microstate is and how we count them. This leads to the statement of the second law of thermodynamics, which as we shall observe has to do with maximizing the entropy of the isolated system.

As the thermodynamic function of state, entropy is simple to understand. Entropy changes of a system are closely joined by heat flow into it. In an infinitesimal reversible method; the heat flowing into the system is the product of the increment in entropy and the temperature. Therefore as heat is not a function of state entropy is.

The Gibbs Approach:

At the starting of the Gibbs approach stands a radical rupture by the Boltzmann program. The object of study for the Boltzmannians is the individual system, comprising of a large however finite number of micro constituents. By contrary, in the Gibbs framework the object of study is a so termed ensemble, an uncountable infinite collection of independent systems that are all regulated by the similar Hamiltonian however distributed over various states. Gibbs states the concept as follows:

We might assume a great number of systems of the similar nature, however differing in the configurations and velocities which they encompass at a given instant, and differing not just infinitesimally, however it might be so as to embrace each and every conceivable combination of configuration and velocities. And here we might set the problem, not to follow a specific system via its succession of configurations, however to find out how the whole number of systems will be distributed among the different conceivable configurations and velocities at any requisite time, if the distribution has been given for some one time.

Ensembles are fictions or mental copies of the one system under consideration; they don't interact by one other, each and every system consists of its own dynamics, and they are not positioned in space and time.

Therefore, it is significant not to confuse ensembles having collections of micro-objects like the molecules of a gas. The ensemble corresponding to a gas made up of 'n' molecules, state, comprises of an infinite number of copies of the whole gas.

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