Thermodynamics statistical pdf

Thermodynamics and Statistical Mechanics will be an invaluable and comprehensive reference manual for research scientists. This text can be used as a complement to existing texts and for supplementary reading.

Offers a fresh perspective on the foundations of statistical thermodynamics Includes a number of new results and novel derivations, and provides an intriguing alternative to existing monographs Simple graphs and figures illustrate the text throughout Logical organization of material An invaluable and comprehensive reference manual for research scientists Can be used as a complement to existing texts and for supplementary reading.

What does statistical physics teach us? In the pages of this slim book, we confront the answers. The reader will discover that where thermodynami cs provi des a 1 arge scal e, macroscopi c theory of the ef fects of temperature on physical systems, statistical mechanics provides the microscopic analysis of these effects which, invariably, are the results of thermal disorder.

A number of systems in nature undergo dramatic changes in aspect and in their properties when subjected to changes in ambient temperature or pres sure, or when electric or magnetic fields are applied.

The ancients already knew that a liquid, a solid, or a gas can represent different states of the same matter. But what is meant by "state"? It is here that the systematic study of magnetic materials has provided one of the best ways of examining this question, which is one of the principal concerns of statistical physics alias "statistical mechanics" and of modern thermodynamics.

Suitable for undergraduates and graduate students, self-study, reference. Crafted and class-tested over many years of teaching, it carefully guides advanced undergraduate and graduate students who are encountering statistical mechanics for the first time through this — sometimes — intimidating subject. The book provides a strong foundation in thermodynamics and the ensemble formalism of statistical mechanics.

An introductory chapter on probability theory is included. Applications include degenerate Fermi systems, Bose-Einstein condensation, cavity radiation, phase transitions, and critical phenomena. The book concludes with a treatment of scaling theories and the renormalization group. In addition, it provides clear descriptions of how to understand the foundational mathematics and physics involved and includes exciting case studies of modern applications of the subject in physics and wider interdisciplinary areas.

Key Features: Presents the subject in a clear and entertaining style which enables the author to take a sophisticated approach whilst remaining accessible Contains contents that have been carefully reviewed with a substantial panel to ensure that coverage is appropriate for a wide range of courses, worldwide Accompanied by volumes on thermodynamics and non-equilibrium statistical mechanics, which can be used in conjunction with this book, on courses which cover both thermodynamics and statistical mechanics.

All Rights Reserved. Thus, according to the second postulate, the thermodynamic equilibrium state of a system is deter- mined by the totality of external parameters and temperature.

Consequently, 1 Equality 1. This conclusion relates any internal parameter Ai to temperature T and external parameters a1 , a2 ,. This equation, written in the symbolic form, is called the equation of state. The number of such equations, naturally, equals the number of internal parameters k. In the simplest case of closed systems,3 if in the capacity of an independent external parameter volume V is accepted, the internal parameter pressure P and internal energy E, conforming to 1.

If from 1. Consequently, thermody- namics is applicable only to ergodic systems. For an ideal gas, using 1. Here the structure of a particles does not interest us, and we will regard that the considered system as consisting of N number of chaoti- cally moving material points interacting among themselves.

Thus, the number of degrees of freedom of the considered system is 3N. In order to have a notion of the number of parti- cles and their dimensions, we quote a known example by Kelvin, according to which the number of H2 O molecules in a glass of water is times the number of glasses of water available in all oceans and seas of the world.

Naturally, it is impossible to describe in detail the state of such a macro- scopic system with a small number of thermodynamic parameters, since these parameters disregard the internal structure of the system. For the complete description of a system, it is necessary to know which particles it consists of, what nature of their interaction is and by which equations their motion is described, i. In conformity with this, in nature two types of systems exist: classical and quantum systems.

We consider these cases separately. Classical systems. The quantities qi t and pi t , i. Each system has its intrinsic phase space. For instance, the phase space of a classical system consisting of N particles represents an abstract 6N -dimensional space.

The phase space can be subdivided into two subspaces: coordinates and impulses. For some systems e. Coordinates qi t and impulses pi t of particles forming a system contin- ually change in conformity with equations of motion 1. This curve is called the phase trajectory. Note that the phase trajectory can be closed, but it cannot intersect or touch itself. This result follows from the principle of determinism of classical mechanics, i. The phase trajectory can be graphically described only for one particle moving in a one-dimensional space.

In this case, the phase space is two-dimensional. In Fig. As seen from Fig. Quantum systems. For quantum systems, i. Indeed, according to the Heisenberg uncertainty principle, the coordinate q and impulse p of a particle cannot be single-valuedly determined simultaneously. One value of the energy of the system En can correspond to each microstate n, or several microstates can correspond to one value of energy, i. Here, we consider the simplest ideal systems, allowing an exact solution.

Ideal gas in a rectangular box. Assume that in a rectangular box with dimensions Lx , Ly , Lz , N non-interacting particles with a mass m are found. If we substitute the value of the wave vector 1. From quantum mechanics, it is known that in such a case each totality of quantum numbers has only one corresponding wave function, i. If the energy of a particle 1. System consisting of harmonic oscillators. Owing to the absence of the interaction between oscillators, expression 1.

Each value ni has one corresponding wave function, i. Inasmuch as the rotators do not interact, the total energy of the system can be determined from expression 1. In fact, if we consider an ideal gas consisting of N diatomic molecules and take into account that each molecule has three translational, one vibra- tional and two rotational degrees of freedom, then the system as a whole has 6N degrees of freedom. From 1. Quasi-classical approximation.

Consider the conditions of quasi- classicity. Condition of quasi-classicity 1. Thus, the free motion of particles is classical when the linear dimensions of the space L in which the motion occurs are much more than the de Broglie wavelength of a particle.

Consider the question of the number of microstates of a system. For quan- tum systems, the number of microstates in the pre-assigned range of energy equals the number of quantum states. To do this, it is required to solve a system of 6N equations 1. Even if the explicit form of the Hamilton function 1. Moreover, knowing the coordinates and impulses of all particles gives no complete information about properties of the system as a whole.

It is associated with the fact that in the behaviour of macrosystems quali- tatively new statistical appropriateness arises.

Such appropriateness bears a probabilistic character and is distinct from mechanical laws. Hence it follows that states of macrosystems should be described by statistical methods. In these methods, the idea is not the exact determination of the microstates of a system but their determination with a certain probability.

Assume that the considered system is a small but macroscopic part of a large system — the thermostat. A microstate of a system interacting with the thermostat chaotically changes over a course of time, and we cannot exactly determine the coordinates q, p of these states in the phase space.

Then the question can be posed in the following form: What is the probability that microstates of a system may be found in a small element of volume dq dp of the phase space? To determine this probability, mentally trace the change in the microstates of a system in the course of a long time interval T.

Assume that over the course of a short time range dt a microstate of the system is found in the element of volume dq dp taken near the point q, p. And this means a solution of system of equations 1. Inasmuch as this system is unsolvable, the immediate calculation of Lt by 1. An advantage of calculation of the statistical mean value 1.

However, as seen from 1. Finding the explicit form of this function is the basic task of statistical physics. Properties of the distribution function. The distribution function ought to satisfy the normalization condition. It is evident that if we integrate the expression 1. Assume that the con- sidered system consists of two macroscopic subsystems.

It is evident that in the process of interaction of these subsystems, basically particles that are found on the boundary, the number of which is much smaller than the total number of particles in the subsystems, participate. Therefore, in a time range which is less than the relaxation time, these subsystems can be con- sidered as self-dependent, i.

Subsystems satisfying this condition are statistically independent. Elements of volume of the phase space of the considered subsystems are denoted by dq 1 dp 1 and dq 2 dp 2 , respectively.

The converse is also true: if the distribution function of the whole system can be presented in the form of the product of the distribu- tion functions of individual subsystems, these subsystems are statistically independent. If we take the logarithm of the equality 1. Liouville theorem. The third property of the distribution function stems from this theorem, according to which the distribution function is con- stant along the phase trajectory, i.

To prove this theorem, mentally keep a watch over microstates of the given subsystem for an extended time and subdivide the time of observation into very small identical ranges. Imag- ine that phase points A1 , A2 , A3 ,. Now assume that each phase point A1 , A2 , A3 ,. It is evident that the multitude of subsystems mentally formed in this way is a multitude of states of identical subsystems with identical Hamilton function and is called the Gibbs statistical ensemble.

The number of subsystems n entering into this ensemble ought to be very large. A microstate of the statistical ensemble, i.

Equation 1. Therefore, in conformity with Liouville theorem 1. Consider the general conclusions stemming from the indicated properties of the distribution function. As it is seen from 1. The explicit form of L q, p is found in classical mechanics. How- ever, using the above-indicated properties of the distribution function, we can determine the general form applicable for any system. According to the third property stemming from the Liouville theorem 1.

Note that the frame of reference can be so connected with the body the system that in the given frame of reference P and M would be equal to zero. Then, in this frame of reference the only mechanical invariant — the total internal energy E q, p — remains, and as a result dependence 1.

And what is the explicit form of this dependence? To this question, a universal answer for any system does not exist. We consider a concrete system. Assume that the considered system consists of several subsystems.

Then, taking into account the condition of additivity of energy 1. In the abstract phase space, 1. Therefore this hypersurface is called the isoenergetic surface. These states are called impossible states. The explicit form of the distribution function for completely isolated clas- sical systems that are found in the thermodynamic equilibrium is determined on the basis of the postulate of statistical physics.

The basic postulate of statistical physics is as follows: If a completely iso- lated system is found in the thermodynamic equilibrium state, the probability of it being found in any possible microstate is identical, i.

For the distribution function, expression 1. The con- stant C is determined from the normalization condition of the distribution function 1. Note that 1. Inasmuch as the number of particles in the system is very large, the purely quantum mechanical description of the system as well as for classical systems is practically impossible.

In fact, for the quantum mechanical description of a system it is necessary to solve 1. The practical impossibility of working with this problem is evident. In nature, a completely isolated system does not exist.

Therefore, the macroscopic system being found in the stationary state is impossible. In statistical physics, it is called the statistical matrix. Impossibility of the stationary state of a macroscopic system also follows from the uncertainty principle for energy. Thus, inasmuch as the quantum mechanical description of the system is impossible, the problem needs to be solved by statistical methods.

To do this, we proceed as follows. Separate one subsystem, which is a small part of a large system. Suppose that the subsystem does not interact with the surroundings.

Here, q are coordinates of all particles of the system, and n is the totality of quantum numbers determining its stationary state. Let the energy of this stationary state be En. And in quantum systems, Wn means the probability of the system that is found in a microstate with energy En , which is determined by the totality of quantum numbers n.

In quantum systems, for the statistical matrix Wnm a theorem anal- ogous to the Liouville theorem can be proved. To do this, using 1. As a result, 1. Therefore, according to 1. In quantum statistics, this conclusion is an analogue of the Liouville theorem 1. From the energetic presentation of the statistical matrix, one more of its property follows. Indeed, as seen from the Liouville equation 1. For the considered system, Lnn is found from quantum mechanics. Naturally, a universal expression for Wn applicable to any system does not exist.

Wn is a conservation quan- tity. Notice that the canonical distribution for systems in the thermostat has the same appearance as 1. As noted above, the energy spectrum of macroscopic systems is continuous. Notice that 1. For an isolated system, quantum states dG falling in the range of energy dE can be considered as possible states. According to the basic postulate of statistical physics, the probability of the system found in any microstate is identical, i.

On the other hand, the probability dW of the system found in any of the states dG ought to be proportional to the number dG. Entropy, as well as energy, is a function of state, i. To do this, suppose that the considered quantum system is found in the thermodynamic equilibrium state. Subdivide this system into a multitude of subsystems. Pass from the distribution over microstates W En to the distribution over energy w E. Even without knowing the explicit form of the distribution function w E , it can be asserted that a subsystem in thermodynamic equilibrium ought to be found most of the time in states close to the mean value of energy E.