Thermodynamics macrostate

Symmetric-matrix valued function, transition state trajectory, colored noise, 208-209 Symmetry rules, linear thermodynamics, macrostates, 11... 

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. 

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... 

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... 

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... 

If one now adds a reservoir with thermodynamic force Xr, then the subsystem macrostate x can change by internal processes A°x, or by exchange with the reservoir, Arx = Axr. Imagining that the transitions occur sequentially,... 

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... 

Of course, depending on the system, the optimum state identified by the second entropy may be the state with zero net transitions, which is just the equilibrium state. So in this sense the nonequilibrium Second Law encompasses Clausius Second Law. The real novelty of the nonequilibrium Second Law is not so much that it deals with the steady state but rather that it invokes the speed of time quantitatively. In this sense it is not restricted to steady-state problems, but can in principle be formulated to include transient and harmonic effects, where the thermodynamic or mechanical driving forces change with time. The concept of transitions in the present law is readily generalized to, for example, transitions between velocity macrostates, which would be called an acceleration, and spontaneous changes in such accelerations would be accompanied by an increase in the corresponding entropy. Even more generally it can be applied to a path of macrostates in time. 

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. 

An NVE system is also referred to as a microcanonical ensemble of particles. In addition to the NVE system, we will encounter NVT (canonical) and NPT (isobaric) systems. Sticking for now to the NVE system, let us imagine that for any given thermodynamic state, or macrostate, the many particles making up... 

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... 

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,... 

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... 

MACROSTATE DISSECTION OF THERMODYNAMIC MONTE CARLO INTEGRALS... 

In Section II we show how characteristic packets and macrostate annealing can be used to efficiently compute thermodynamic integrals over complicated potentials, and in Section III we show how the same ideas can be used to efficiently compute intermacrostate transition rates. In Section IV we show how this approach can be adapted for analyzing macromolecular integrals and demonstrate it on the pentapeptide Met-enkephalin. We summarize and discuss the next steps to be taken in Section V. For simplicity we consider the evaluation of the conformational partition function Zc, but the method also applies to averages such as Eqs. (1.5). Although we focus on proteins, it should be possible to apply a similar approach to problems... 

Macrostate thermodynamic parameters are computed as integrals over pBi . The macrostate conformational free energy Fx is... 

At biological temperature (—0.6 kcal/mol) pG x 0.1, and averages over G and a handful of other macrostates are adequate for computing thermodynamics properties. For simplicity, we focus on finding G, but the same... 

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