File Name: thermodynamics and statistical physics .zip
In physics , statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic behavior of nature from the behavior of such ensembles. Statistical mechanics arose out of the development of classical thermodynamics , a field for which it was successful in explaining macroscopic physical properties such as temperature , pressure , heat capacity , in terms of microscopic parameters that fluctuate about average values, characterized by probability distributions.
In physics , statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities.
It does not assume or postulate any natural laws, but explains the macroscopic behavior of nature from the behavior of such ensembles. Statistical mechanics arose out of the development of classical thermodynamics , a field for which it was successful in explaining macroscopic physical properties such as temperature , pressure , heat capacity , in terms of microscopic parameters that fluctuate about average values, characterized by probability distributions.
This established the field of statistical thermodynamics and statistical physics. The founding of the field of statistical mechanics is generally credited to Austrian physicist Ludwig Boltzmann , who developed the fundamental interpretation of entropy in terms of a collection of microstates, to Scottish physicist James Clerk Maxwell , who developed models of probability distribution of such states, and to American Josiah Willard Gibbs , who coined the name of the field in While classical thermodynamics is primarily concerned with thermodynamic equilibrium , statistical mechanics has been applied in non-equilibrium statistical mechanics to the issues of microscopically modeling the speed of irreversible processes that are driven by imbalances.
Examples of such processes include chemical reactions or flows of particles and heat. The fluctuation—dissipation theorem is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles. In physics, two types of mechanics are usually examined: classical mechanics and quantum mechanics. For both types of mechanics, the standard mathematical approach is to consider two concepts:.
Using these two concepts, the state at any other time, past or future, can in principle be calculated. There is however a disconnection between these laws and everyday life experiences, as we do not find it necessary nor even theoretically possible to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale for example, when performing a chemical reaction.
Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in.
Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the statistical ensemble , which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a probability distribution over all possible states of the system.
In classical statistical mechanics, the ensemble is a probability distribution over phase points as opposed to a single phase point in ordinary mechanics , usually represented as a distribution in a phase space with canonical coordinates.
In quantum statistical mechanics, the ensemble is a probability distribution over pure states, [note 1] and can be compactly summarized as a density matrix. As is usual for probabilities, the ensemble can be interpreted in different ways: . These two meanings are equivalent for many purposes, and will be used interchangeably in this article. However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion.
Thus, the ensemble itself the probability distribution over states also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the Liouville equation classical mechanics or the von Neumann equation quantum mechanics. These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state.
One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as equilibrium ensembles and their condition is known as statistical equilibrium.
Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state. The primary goal of statistical thermodynamics also known as equilibrium statistical mechanics is to derive the classical thermodynamics of materials in terms of the properties of their constituent particles and the interactions between them.
In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in thermodynamic equilibrium , and the microscopic behaviours and motions occurring inside the material. Whereas statistical mechanics proper involves dynamics, here the attention is focussed on statistical equilibrium steady state.
Statistical equilibrium does not mean that the particles have stopped moving mechanical equilibrium , rather, only that the ensemble is not evolving. A sufficient but not necessary condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties total energy, total particle numbers, etc. A common approach found in many textbooks is to take the equal a priori probability postulate.
The equal a priori probability postulate therefore provides a motivation for the microcanonical ensemble described below. There are various arguments in favour of the equal a priori probability postulate:. Other fundamental postulates for statistical mechanics have also been proposed.
There are three equilibrium ensembles with a simple form that can be defined for any isolated system bounded inside a finite volume. In the macroscopic limit defined below they all correspond to classical thermodynamics.
For systems containing many particles the thermodynamic limit , all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used.
In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.
Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' macroscopic observables can be extracted from the characteristic state function.
Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system.
While some hypothetical systems have been exactly solved, the most general and realistic case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.
One approximate approach that is particularly well suited to computers is the Monte Carlo method , which examines just a few of the possible states of the system, with the states chosen randomly with a fair weight.
As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained.
As more and more random samples are included, the errors are reduced to an arbitrarily low level. There are many physical phenomena of interest that involve quasi-thermodynamic processes out of equilibrium, for example:. All of these processes occur over time with characteristic rates, and these rates are of importance for engineering.
The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium. In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as Liouville's equation or its quantum equivalent, the von Neumann equation.
These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. Unfortunately, these ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain.
Moreover, the ensemble evolution equations are fully reversible and do not destroy information the ensemble's Gibbs entropy is preserved. In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics. Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored.
A few approaches are described in the following subsections. One approach to non-equilibrium statistical mechanics is to incorporate stochastic random behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate aside from hypothetical situations involving black holes , a system cannot in itself cause loss of information , the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment.
These correlations appear as chaotic or pseudorandom influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier. The Boltzmann transport equation and related approaches are important tools in non-equilibrium statistical mechanics due to their extreme simplicity.
These approximations work well in systems where the "interesting" information is immediately after just one collision scrambled up into subtle correlations, which essentially restricts them to rarefied gases. The Boltzmann transport equation has been found to be very useful in simulations of electron transport in lightly doped semiconductors in transistors , where the electrons are indeed analogous to a rarefied gas.
Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in linear response theory. A remarkable result, as formalized by the fluctuation—dissipation theorem , is that the response of a system when near equilibrium is precisely related to the fluctuations that occur when the system is in total equilibrium.
Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or "know" how it came to be away from equilibrium. This provides an indirect avenue for obtaining numbers such as ohmic conductivity and thermal conductivity by extracting results from equilibrium statistical mechanics.
Since equilibrium statistical mechanics is mathematically well defined and in some cases more amenable for calculations, the fluctuation—dissipation connection can be a convenient shortcut for calculations in near-equilibrium statistical mechanics. An advanced approach uses a combination of stochastic methods and linear response theory. As an example, one approach to compute quantum coherence effects weak localization , conductance fluctuations in the conductance of an electronic system is the use of the Green—Kubo relations, with the inclusion of stochastic dephasing by interactions between various electrons by use of the Keldysh method.
The ensemble formalism also can be used to analyze general mechanical systems with uncertainty in knowledge about the state of a system. Ensembles are also used in:. In , Swiss physicist and mathematician Daniel Bernoulli published Hydrodynamica which laid the basis for the kinetic theory of gases. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion.
In , after reading a paper on the diffusion of molecules by Rudolf Clausius , Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range.
Statistical mechanics was initiated in the s with the work of Boltzmann, much of which was collectively published in his Lectures on Gas Theory.
Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his H -theorem. The term "statistical mechanics" was coined by the American mathematical physicist J. Willard Gibbs in From Wikipedia, the free encyclopedia. Physics of large number of particles' statistical behavior. Particle statistics. Thermodynamic ensembles. Debye Einstein Ising Potts. Main articles: Mechanics and Statistical ensemble.
Main articles: Microcanonical ensemble , Canonical ensemble , and Grand canonical ensemble. Number of microstates. Canonical partition function. Grand partition function. Boltzmann entropy. Helmholtz free energy. Grand potential. Main article: Monte Carlo method. See also: Non-equilibrium thermodynamics.
In this post we will work on the derivation of thermal conductivity formula first, then we will find the dimension of thermal conductivity as well. Thermal Stress is the stress caused due to the change in temperature. Topic 5: Electricity and Magnetism Quiz — 14 sub-questions. In a similar spirit, this paper illustrates some of the physics principles that become apparent when an IR camera is used to observe the atmosphere. Thermal Physics, Schroeder Now, it is true. Topic 9 - Motion in Fields.
The textbooks that I have consulted most frequently while developing course material are: Fundamentals of Statistical and Thermal Physics: F. Reif (McGraw-Hill.
Building from first principles, it gives a transparent explanation of the physical behaviour of equilibrium thermodynamic systems, and it presents a comprehensive, self-contained account of the modern mathematical and computational techniques of statistical mechanics. Dr Attard is a well-known researcher in statistical mechanics who has made significant contributions to this field. His book offers a fresh perspective on the foundations of statistical thermodynamics. It includes a number of new results and novel derivations, and provides an intriguing alternative to existing monographs. Especially of note are the simple graphs and figures that illustrate the text throughout and the logical organization of the material. 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.
Applied Thermodynamics for Engineers by William D. Ennis, The basic paradoxes of statistical classical physics and quantum mechanics by Oleg Kupervasser, , pages, 2.
The Temperature of solid is externally maintained so that heat can flow from high temp to low temp. Thermodynamics and heat power 6th. Measure of Heat. Isothermal and adiabatic processes. The rise in temperature of a substance when work is done is well known. The energy processes that convert heat energy from available sources such as chemical fuels into mechanical work are the major concern of this science.
The lecture notes are from an earlier version of this course, but still correspond to the topics covered in this version. Don't show me this again. This is one of over 2, courses on OCW.
Стратмору, разумеется, это было хорошо известно, но даже когда Сьюзан порывалась уйти через главный выход, он не обмолвился об этом ни единым словом. Он не мог пока ее отпустить - время еще не пришло. И размышлял о том, что должен ей сказать, чтобы убедить остаться.
Мужчина достал мобильник, сказал несколько слов и выключил телефон. - Veinte minutos, - сказал. -Двадцать минут? - переспросил Беккер. - Yel autobus. Охранник пожал плечами.
Вирус. - Никакого вируса. Выслушай меня внимательно, - попросил Стратмор. Сьюзан была ошеломлена.
У меня галлюцинация. Когда двери автобуса открылись, молодые люди быстро вскочили внутрь. Беккер напряг зрение.
Вопрос насколько. уступил место другому - с какой целью?. У Хейла не было мотивов для вторжения в ее компьютер.
А зачем это нам? - спросила Сьюзан. - В этом нет никакого смысла. Стратмор встал и начал расхаживать по кабинету, не спуская при этом глаз с двери. - Несколько недель назад, когда я прослышал о том, что Танкадо предложил выставить Цифровую крепость на аукцион, я вынужден был признать, что он настроен весьма серьезно. Я понимал, что если он продаст свой алгоритм японской компании, производящей программное обеспечение, мы погибли, поэтому мне нужно было придумать, как его остановить.
Ошибиться было невозможно. Это мощное тело принадлежало Грегу Хейлу. ГЛАВА 58 - Меган - девушка моего друга Эдуардо! - крикнул панк Беккеру. -Держись от нее подальше.
Стратмор выключил телефон и сунул его за пояс. - Твоя очередь, Грег, - сказал. ГЛАВА 81 С мутными слезящимися глазами Беккер стоял возле телефонной будки в зале аэровокзала. Несмотря на непрекращающееся жжение и тошноту, он пришел в хорошее расположение духа.
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