Thermodynamic probability defined

A quantitative way of dealing with the degree of disorder in a system is to define something called the thermodynamic probability Q. which counts the number of ways in which a particular state can come about. Thus situations we characterize as relatively disordered can come about in more ways than a relatively ordered state, just as an unordered deck of cards compared to a deck arranged by suits. 

If the polymer system was able to exist in an equilibrium state only, then a strictly defined correlation between (a, ph) and (a, ph) would exist in particular conditions, according to minimum of free energy of system formation. Consequently, there would occur only one temperature at which process initiation is thermodynamically probable. In rare ca.ses there may occur different correlations between ( ph, a) and ( ph, a ), which display one and the same value of free energy minimum of system formation. 

Finally, equilibrium processes can be defined as processes between and passing states that all have the same thermodynamic probability. On the one hand, these processes proceed without driving forces on the other hand, and this is inconsistent and unrealistic, there is no incentive for the process to proceed. These imaginary processes function only to establish the minimum amount of work required, or the maximum amount of work available, in proceeding from one state to the other. 

These ideas were expressed mathematically by L. Boltzmann and J. W. Gibbs in terms of a quantity ft, called the thermodynamic probability and defined as the number of ways that microscopic particles can be distributed among the states accessible to them. It is given by the general formula... 

While the residue stability constants are purely thermodynamic quantities defined for all residues, the protection factors also contain non-thermodynamic contributions and are defined only for a subset of residues. For example, proline residues lack the amide group and therefore are not included. From a statistical standpoint, the protection factor for any given residue] can be defined as the ratio of the sum of the probabilities of the states in which residue j is closed, to the sum of the probabilities of the states in which residue] is open ... 

In order to determine the statistical thermodynamic probabilities and entropies for the conformational energy surface, a set of "dots" is plotted indicating the angular values of the set of conformers which define the surface. The joystick curser control 1s used to select the set of conformers which occupy a given low energy region. The chosen "dots" are replaced by "asterisks" (to avoid duplication) and the probability and entropy terms are tabulated. Tables of probabilities and entropies may also be produced. 

The thickness of the amorphous regions is probably defined by the minimum distance, two independently growing nuclei can have to each other. This value is obviously not influenced by the temperature. Above 235 °C, la seems to increase at expense of the crystalline regions. The sample is in a partially molten state, which leads to the existence of amorphous islands. It could be shown that these islands are filled with smaller lamellae, when the temperature is decreased so far that they can become thermodynamically stable. 

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 thermodynamic probability of a system is defined as the ratio of the probability of an actual state to one of the same total energy and volume in which the molecules are completely ordered. This suggests that entropy is a function of probability (P) that is. 

The easiest way to observe the connection between entropy and disorder is to study ideal gases. First, one defines the disorder as the thermodynamic probability W. the number of ways a system can be arranged (in contrast to the normally chosen probability that would be a fraction). Next, one notices that W is multiplicative. The combined probability of systems one and two is W = Wi x W2. The entropy, in contrast, is extensive, re. S = Si + S2. Any connection between the two must thus be ... 

The function to(j) determines the probability of a complicated event, meaning that the RW trajectory is also a SAW trajectory at the same time and with it last step it falls into one of 2 equiprobable cells Mp s) one can also say that it realizes the state Afp(s). This implies that it is numerically equal to the part of those SAW trajectories from the whole quantity of RW trajectories which realize the state Mp(s). The quantity L(s) of such trajectories define the thermodynamical probability of realization of the state Mp(s)... 

The state of the previously defined collection is constantly changing, such that over time the collection is distributed thereby its objects may find themselves in different states. The number of complexions is the number of distributions of objects between the different states that they are likely to take. Of all the possible distributions, there is one that is the maximum of the number of complexions. The Boltzmann principle states that the number of complexions corresponding to the most probably distribution type is almost equal to the total number of complexions and vice versa. The state of the collection therefore always corresponds to the maximum of complexions, which we call the thermodynamic probability or dominant probability. 

If a macroscopic system such as a gas is at thermodynamic equilibrium, its entropy has a well-defined constant value. However, the molecules are moving and occupy many different molecular states without changing the macroscopic state or the value of the entropy. A single macroscopic state and a single value of the entropy must correspond to many microscopic states of the system. We define the thermodynamic probability... 

We consider a two state system, state A and state B. A state is defined as a domain in phase space that is (at least) in local equilibrium since thermodynamic variables are assigned to it. We assume that A or B are described by a local canonical ensemble. There are no dark or hidden states and the probability of the system to be in either A or in B is one. A phenomenological rate equation that describes the transitions between A and B is... 

There is a fundamental relationship between d-dimensional PCA and d + 1)-dimensional Ising spin models. The simplest way to make the connection is to think of the successive temporal layers of the PCA as successive hyper-planes of the next higher-dimensional spatial lattice. Because the PCA rules (at least the set of PCA rules that we will be dealing with) are (1) Markovian (i.e. the probability of a state at time t + T depends only on a set of states at time t, and (2) local, one can always define a Hamiltonian on the higher-dimensioned spatial lattice such that the thermodynamic weight of a configuration 5j,( is equal to the probability of a corresponding space-time history Si t). ... 

In the first chapter, we defined the nature of a solid in terms of its building blocks plus its structure and symmetry. In the second chapter, we defined how structures of solids are determined. In this chapter, we will examine how the solid actually occurs in Nature. Consider that a solid is made up of atoms or ions that are held together by covalent/ionic forces. It is axiomatic that atoms cannot be piled together and forced to form a periodic structure without mistakes being made. The 2nd Law of Thermodynamics demands this. Such mistakes seriously affect the overall properties of the solid. Thus, defeets in the lattice are probably the most important aspect of the solid state since it is impossible to avoid defects at the atomistic level. Two factors are involved ... 

Statistical thermodynamics is based on a statistical interpretation of how atoms and molecules behave. This statistical nature arises because we have so many atoms and molecules in systems and because matter is intrinsically defined based on probabilities, which is the crux of all quantum mechanics. Rather than delve into the great details of statistical thermodynamics, which would far exceed the scope of this text, we will present its foundations only. 

That this should be so is a corollary of the Second Law of Thermodynamics which is concerned essentially with probabilities, and with the tendency for ordered systems to become disordered a measure of the degree of disorder of a system being provided by its entropy, S. In seeking their most stable condition, systems tend towards minimum energy (actually enthalpy, H) and maximum entropy (disorder or randomness), a measure of their relative stability must thus embrace a compromise between H and S, and is provided by the Gibb s free energy, G, which is defined by,... 

The most broadly recognized theorem of chemical thermodynamics is probably the phase rule derived by Gibbs in 1875 (see Guggenheim, 1967 Denbigh, 1971). Gibbs phase rule defines the number of pieces of information needed to determine the state, but not the extent, of a chemical system at equilibrium. The result is the number of degrees of freedom Np possessed by the system. 

Boltzmann, following Clausius, considered entropy to be defined only to an arbitrary constant, and related the difference in entropy between two states of a system to their relative probability. An enormous advance was made by Planck who proposed to determine the absolute entropy as a quantity, which, for every realizable system, must always be positive (third law of thermodynamics). He related this absolute entropy, not to the probability of a system, but to the total number of its possibilities. This view of Planck has been the basis of all recent efforts to find the statistical basis of thermodynamics, and while these have led to many differences of opinion, and of interpretation, we believe it is now possible to derive the second law of thermodynamics in an exact form and to obtain... 

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