Thermodynamics microstate

Due to individal particles being undistmguishable mutually by means of macroscopic thermodynamics, microstates of a given macrostate are undidisguishable too State space of A can be three-dimensional or general configuration space, impulse or the whole phase space... 

The entropy change is significant, and the net change is positive (this would correspond to the presence of more thermodynamic microstates in the product compared to the reactant). In this case the maximum thermodynamic efficiency would increase with temperature. 

If a confined fluid is thermodynamically open to a bulk reservoir, its exposure to a shear strain generally gives rise to an apparent multiplicity of microstates all compatible with a unique macrostate of the fluid. To illustrate the associated problem, consider the normal stress which can be computed for various substrate separations in grand canonical ensemble Monte Carlo simulations. A typical curve, plotted in Fig. 16, shows the oscillatory decay discussed in Sec. IV A 2. Suppose that instead... 

In equation (1.17), S is entropy, k is a constant known as the Boltzmann constant, and W is the thermodynamic probability. In Chapter 10 we will see how to calculate W. For now, it is sufficient to know that it is equal to the number of arrangements or microstates that a molecule can be in for a particular macrostate. Macrostates with many microstates are those of high probability. Hence, the name thermodynamic probability for W. But macrostates with many microstates are states of high disorder. Thus, on a molecular basis, W, and hence 5, is a measure of the disorder in the system. We will wait for the second law of thermodynamics to make quantitative calculations of AS, the change in S, at which time we will verify the relationship between entropy and disorder. For example, we will show that... 

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. 

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... 

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 ... 

Now suppose that some external constraint on the system, is removed. New microstates, previously inaccessible, become available and transition into these new states may occur. As a consequence the number of microstates among which transitions occur, increases to the maximum permitted by the remaining constraints. This statement is strikingly reminiscent of the entropy postulate of thermodynamics, according to which the entropy increases to the maximum permitted by the imposed constraints. It appears that entropy may be identified with the number of allowed microstates consistent with the imposed macroscopic constraints. 

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. 

For classical systems the microstates are not discrete and the number of possible states for a fixed NVE ensemble is in general not finite. To see this imagine a system of a single particle (N = 1) traveling in an otherwise empty box of volume V. There are no external force fields acting on the particle so its total energy is E = mv2. The particle could be found in any location within the box, and its velocity could be directed in any direction without changing the thermodynamic macrostate defined by the fixed values of N, V, and E. To apply ensemble theory to classical systems Q(N, V, E) is defined as the (appropriately scaled) total volume accessible by the state variables of position and momentum accessible by the particles in the system. 

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 ... 

Again, the term macrostate refers to the thermodynamic state of the composite system, defined by the variables N, E, and V2, E2. A more probable macrostate will be one that corresponds to more possible microstates... 

To summarize, we have shown that a specific physical interpretation of the intensive variables governed by Equation (1.1) - temperature, pressure, and chemical potential - arises from the assumption that systems move to thermodynamic macrostates that maximize the number of accessible microstates. This is our first application of the famous second law of thermodynamics, which, as is implicit in the above derivations, is stated as the entropy of a closed system never decreases. It is worth noting that our interpretation of the intensive thermodynamic variables... 

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. 

We may be tempted to call the group of particles existing in a microstate as a subsystem. But the word system has such a connotation which has led scientists to use, instead, a term Ensemble basically meaning a group. In classical thermodynamics, we say that (as per zeroeth law) two systems in thermal equilibrium with a third system are in thermal equilibrium with each other and the same can be attributed to subsystems, but not to ensembles . 

The unstable degrees of freedom determine the number of various allowed microstates that are responsible for creating the given macrostate. This is namely the number of the microstates, or their thermodynamic probabihty Qj , which determines a total of entropy S of the system. According to the Boltzmann formula,... 

Goldbeck RA, Esquerra RM, Holt JM, Ackers GK, Kliger DS. The molecular code for hemoglobin allostery revealed by linking the thermodynamics and kinetics of quaternary structural change. 1. Microstate linear free energy relations. Biochemistry 2004 43 12048-12064. 

There is a third possibility. There could be a purely dynamic transition at or near To, in which ergodicity is broken, but all thermodynamic variables remain continuous, so that no thermodynamic transition occurs. This would mean that below a critical temperature in the vicinity of Tg, the system is kinetically prevented from exploring all microstates that are thermodynamically allowed, but gets permanently locked into a finite subset of these... 

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. 

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