Macrostates thermodynamic probability

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... 

Where k is Boltzmann constant and Q the weight of configuration (thermodynamic probability). Q is defined as the number of microstates forming a macrostate. With... 

The standard construction requires that the thermodynamic function of the macrostate it will be written by combining the statistical information contained within the thermodynamic probability (1.157) with the Langrange constraints of particle and energy conservation... 

Let us denote the cardinality of such a class as thermodynamic probability P of a given macrostate. 

Then the Boltzman definition (11) of the physical (thermodynamic) entropy S of. A (of, the whole volume V, or, of a given state space) in a macrostate with thermodynamic probability P [could be non-equilibrium too, composed by (equilibrium) subsystems A in states 0, i 1,2, not interacting mutually (at the same temperature 0, or also, at various... 

Since molecules can occupy various states without changing the macroscopic state of the system of which they are a part, it is apparent that many microstates of a macroscopic system correspond to one macroscopic state. We denote the number of microstates corresponding to a given macrostate by Q. The quantity is sometimes called the thermodynamic probability of the macrostate. The thermodynamic probability is a measure of lack of information about the microstate of the system for a particular macrostate. A large value corresponds to a small amount of information, and a value of unity corresponds to knowledge that the system is in a specific microstate. 

The so-called thermodynamic probability of a macrostate s statistical weights is used in statistical physics. The thermodynamic probability W of a system macrostate is determined by a number of various microstates that can assure the given maCTOstate. 

Boltzmann showed that a system s entropy is proportional to the natural logarithm of the number of possible microstates corresponding to the given macrostate, i.e., to the thermodynamic probability. The proportionality factor is k, a value later called the Boltzmann constant. Therefore, entropy can be expressed as... 

The present theory can be placed in some sort of perspective by dividing the nonequilibrium field into thermodynamics and statistical mechanics. As will become clearer later, the division between the two is fuzzy, but for the present purposes nonequilibrium thermodynamics will be considered that phenomenological theory that takes the existence of the transport coefficients and laws as axiomatic. Nonequilibrium statistical mechanics will be taken to be that field that deals with molecular-level (i.e., phase space) quantities such as probabilities and time correlation functions. The probability, fluctuations, and evolution of macrostates belong to the overlap of the two fields. 

In Sections IVA, VA, and VI the nonequilibrium probability distribution is given in phase space for steady-state thermodynamic flows, mechanical work, and quantum systems, respectively. (The second entropy derived in Section II gives the probability of fluctuations in macrostates, and as such it represents the nonequilibrium analogue of thermodynamic fluctuation theory.) The present phase space distribution differs from the Yamada-Kawasaki distribution in that... 

The generic case is a subsystem with phase function x(T) that can be exchanged with a reservoir that imposes a thermodynamic force Xr. (The circumflex denoting a function of phase space will usually be dropped, since the argument T distinguishes the function from the macrostate label x.) This case includes the standard equilibrium systems as well as nonequilibrium systems in steady flux. The probability of a state T is the exponential of the associated entropy, which is the total entropy. However, as usual it is assumed (it can be shown) [9] that the... 

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. 

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... 

Furthermore, the thermodynamic function of the macrostate is formed, using the last form of Fermi-Dirac probability to look like... 

In statistic thermodynamics the entropy of the equilibrious system equals the logarithm of the probability of its definite macrostate ... 

This statement is easy to understand now in terms of macrostates and multiplicity. For a large system of 10 particles, probability dictates that the material will be in the most likely macrostate or one very close to this state. Say, for example, the gas in a room spontaneously moves to occupy a volume representing 50% of the room s volume the probability of this occurring will be massively less likely to occur than the most probable state (filling the entire room). Such scenarios are so unlikely that we can state that they never happen. The overwhelming likelihood that a system will remain in or close to the most probable state (in equilibrium) means that in the thermodynamic limit, entropy will not decrease spontaneously and can only increase. 

The minima are separated by the maximum at the transition state, Fts = F T qts) If g is indeed the relevant degree of freedom and the system resides in the local minimum at A, the free-energy difference =Fts — Fa is the kinetic barrier the system has to climb to reach the transition state. Only after reaching the ensemble of states belonging to the thermodynamically unstable transition macrostate, the system is with a certain probability p able to enter the ordered stable equilibrium state at B. With the probability 1 — p it can also happen that the system returns to A again. However, since in the example shown in Fig. 2.1 Fb < Fa, the system will macroscopically reside in the thermodynamic... 

This has been accomplished not only for NVB systems, but for NVT,NPT, and ixVT systems among others. Indeed, for any system in an equilibrium macroscopic state, statistical thermodynamics focuses on the determination of the probabilities of all the microscopic states that correspond to the equilibrium macrostate. It also focuses on the enumeration of these microscopic states. With the information of how many microscopic states correspond to a macroscopic one and of what their probabilities are, the thermodynamic state and behavior of the system can be completely determined. 

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