Statistical thermodynamics microstates

Self-organization seems to be counterintuitive, since the order that is generated challenges the paradigm of increasing disorder based on the second law of thermodynamics. In statistical thermodynamics, entropy is the number of possible microstates for a macroscopic state. Since, in an ordered state, the number of possible microstates is smaller than for a more disordered state, it follows that a self-organized system has a lower entropy. However, the two need not contradict each other it is possible to reduce the entropy in a part of a system while it increases in another. A few of the system s macroscopic degrees of freedom can become more ordered at the expense of microscopic disorder. This is valid even for isolated, closed systems. Eurthermore, in an open system, the entropy production can be transferred to the environment, so that here even the overall entropy in the entire system can be reduced. 

Monte Carlo heat flow simulation, 69-70 nonequilibrium statistical mechanics, microstate transitions, 44 46 nonequilibrium thermodynamics, 7 time-dependent mechanical work, 52-53 transition probability, 53-57 Angular momentum, one- vs. three-photon... 

Now we introduce a fundamental postulate of statistical thermodynamics at a given Nx, Vx, and Ex, system 1 is equally likely to be in any one of its 121 microstates similarly system 2 is equally likely to be in any one of its 122 microstates (more on this assumption later). The combined system, consisting of systems 1 and 2, has associated with it a total partition function 120(.Ei, E2), which represents the total number of possible microstates. The number 120(.Ei, E2) may be expressed as the multiplication ... 

Matters are made up of small particles such as molecules and atoms. Thermodynamic laws have been postulated and inferred without looking into the micro-properties or microstates within the systems. A branch of thermodynamics has evolved, which tries to interpret thermodynamic properties based on the properties of micro constituent of the system. This branch is called the Statistical Thermodynamics. An offshoot is the Nuclear Thermodynamics , where matter is treated as another form of energy and role of atomic and subatomic particle forms are studied in determining thermodynamic properties. 

The third law, like the two laws that precede it, is a macroscopic law based on experimental measurements. It is consistent with the microscopic interpretation of the entropy presented in Section 13.2. From quantum mechanics and statistical thermodynamics, we know that the number of microstates available to a substance at equilibrium falls rapidly toward one as the temperature approaches absolute zero. Therefore, the absolute entropy defined as In O should approach zero. The third law states that the entropy of a substance in its equilibrium state approaches zero at 0 K. In practice, equilibrium may be difficult to achieve at low temperatures, because particle motion becomes very slow. In solid CO, molecules remain randomly oriented (CO or OC) as the crystal is cooled, even though in the equilibrium state at low temperatures, each molecule would have a definite orientation. Because a molecule reorients slowly at low temperatures, such a crystal may not reach its equilibrium state in a measurable period. A nonzero entropy measured at low temperatures indicates that the system is not in equilibrium. 

As you no doubt see, there would be such a staggeringly large number of microstates that taking individual snapshots of all of them is not feasible. Because we are examining such a large number of particles, however, we can use the tools of statistics and probability to determine the total number of microstates for the thermodynamic state. (That is where the statistical part of the name statistical thermodynamics comes in.) Each thermodynamic state has a characteristic number of microstates associated with it, and we will use the symbol W for that number. 

One of the main concerns in statistical thermodynamics is to describe how reversible microscopic equations of motion produce irreversible macroscopic behavior. One can stndy the macroscopic behavior of macroscopic systems by considering just one of the very large numbers of microstates that can satisfy the macroscopic properties and then solving the equations of motion for this single microscopic representative trajectory. 

All macroscopic observables are obtainable from the distribution of microscopic states (henceforth, microstates) of elements that obey mechanics. Strictly speaking, this is the most basic assumption of statistical thermodynamics. However, to elucidate macroscopic phenomena, it is not necessary to know the true distribution of microstates of the system. Boltzmann introduced the concept of orthodic ensembles that are compatible with thermodynamics. In practice, an orthodic ensemble is established hy demonstrating that it is compatible with the laws of mechanics and with the laws of thermodynamics. Gibbs demonstrated that the canonical distribution fimction... 

A chemical system will adopt the arrangement that offers the maximum statistical probability of microstates. This description of entropy is known as statistical thermodynamics. 

The concept of entropy was developed so chemists could understand the concept of spontaneity in a chemical system. Entropy is a thermodynamic property that is often associated with the extent of randomness or disorder in a chemical system. In general if a system becomes more spread out, or more random, the system s entropy increases. This is a simplistic view, and a deeper understanding of entropy is derived from Ludwig Boltzmann s molecular interpretation of entropy. He used statistical thermodynamics (which uses statistics and probability) to link the microscopic world (individual particles) and the macroscopic world (bulk samples of particles). The connection between the number of microstates (arrangements) and its entropy is expressed in the Boltzmann equation, S = kin W, where W is the number of microstates and k is the Boltzmann constant, 1.38 x 10 JK L... 

In statistical thermodynamics, a system with interacting particles is depicted with the canonical ensemble that describes a collection of a large number of macroscopic systems under identical conditions (for instance, N particles in a volume V at temperature T). In each system, laws that describe interactions between molecules are identical. They differ by the coordinates of each particular molecule corresponding to a microstate. The static picture of the canonical ensemble is equivalent to the development of a system over time [10,14]. In other words, the measurement of a macroscopic property reflects a succession of microstates. Thus, the measured property corresponds to a time-averaged mean value and thermodynamic equilibrium corresponds to the most probable macroscopic state. 

A fundamentally important concept of statistical thermodynamics is the microstate of a system. We define a microstate of a system by the values of the positions and velocities of all the N particles. We can concisely describe a microstate with a 6A-dimensional vector... 

A fundamental hypothesis in statistical thermodynamics is that the two averages are equal. This is the ergodic hypothesis. The physical implication of the ergodic hypothesis is that any system afforded with infinite time will visit all the points of phase space with a frequency proportional to their probability density. In other words, in a single trajectory the system spends an amount of time at each microstate that is proportional to its probability. The ergodic hypothesis is still a hypothesis because there has been no formal proof of its truth. 

It is the ergodic hypothesis that allowed Gibbs to shift attention from trajectories to probabihties in phase space. Instead of considering orbits of microstate chains crossing the phase space in time, one can envision the phase space as a continuum with a position-dependent density. Because the latter can be determined more readily than the former, statistical thermodynamics can be employed to connect microscopic to macroscopic states. 

We can show that the thermodynamic and statistical entropies are equivalent by examining the isothermal expansion of an ideal gas. We have seen that the thermodynamic entropy of an ideal gas increases when it expands isothermally (Eq. 3). If we suppose that the number of microstates available to a single molecule is proportional to the volume available to it, we can write W = constant X V. For N molecules, the number of microstates is proportional to the Nth power of the volume ... 

Doubling the number of molecules increases the number of microstates from W to W2, and so the entropy changes from k In W to k In W2, or 2k In W. Therefore, the statistical entropy, like the thermodynamic entropy, is an extensive property. 

Nonequilibrium statistical mechanics Green-Kubo theory, 43-44 microstate transitions, 44-51 adiabatic evolution, 44—46 forward and reverse transitions, 47-51 stationary steady-state probability, 47 stochastic transition, 464-7 steady-state probability distribution, 39—43 Nonequilibrium thermodynamics second law of basic principles, 2-3 future research issues, 81-84 heat flow ... 

Until now we assumed that we have the maximum information on the many-particle system. Now we will consider a large many-body system in the so-called thermodynamic limit (N- °o, V—> >, n = NIV finite) that means a macroscopic system. Because of the (unavoidable) interaction of the macroscopic many-particle system with the environment, the information of the microstate is not available, and the quantum-mechanical description is to be replaced by the quantum-statistical description. Thus, the state is characterized by the density operator p with the normalization... 

Similarly, if one is interested in a macroscopic thermodynamic state (i.e., a subset of microstates that corresponds to a macroscopically observable system with bxed mass, volume, and energy), then the corresponding entropy for the thermodynamic state is computed from the number of microstates compatible with the particular macrostate. All of the basic formulae of macroscopic thermodynamics can be obtained from Boltzmann s definition of entropy and a few basic postulates regarding the statistical behavior of ensembles of large numbers of particles. Most notably for our purposes, it is postulated that the probability of a thermodynamic state of a closed isolated system is proportional to 2, the number of associated microstates. As a consequence, closed isolated systems move naturally from thermodynamic states of lower 2 to higher 2. In fact for systems composed of many particles, the likelihood of 2 ever decreasing with time is vanishingly small and the second law of thermodynamics is immediately apparent. 

In the molecular statistical analysis, Boltzmann defined the entropy S in any thermodynamic state as S = In fl, where O is the number of microstates available to the system in that same thermodynamic state. This equation is used for qualitative interpretations of entropy changes. It shows that any process that increases D will increase S, and any process that decreases O will decrease S. 

In comparison with the equilibrium thermodynamics, the system in the equilibrium statistical mechanics is described by two additional elements the microstates of the system and the... 

In the equilibrium statistical mechanics, the unknown probabilities of microstates p, are found from the second part of the second law of thermodynamics, i.e., from the constrained extremum of the thermodynamic potential (Eq. (29)) as a function of the variables (pv pw) under the condition that the variables (pv. .., pw) satisfy Eq. (27). Moreover, it is supposed that the value of the entropy in the i th microstate of the system is a function of the probability pt of this microstate, i.e., =Sf=Sf(pf). Then to determine the unknown probabilities [pt] at... 

As thermodynamics required postulates or laws, so does statistical mechanics. Gibbs postulates which define statistical mechanics are (1) Thermodynamic quantities can be mapped onto averages over all possible microstates consistent with the few macrosopic parameters required to specify the state of the system (here, NVE). (2) We construct the averages using an ensemble . An ensemble is a collection of systems identical on the macroscopic level but different on the microscopic level. (3) The ensemble members obey the principle of equal a priori probability . That is, no one ensemble member is more important or probable than another. 

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