Microscopic state function

Any variable that depends on the positions and velocities of the particles is determined by the state of the system and is called a microscopic state function or a mechanical state function. The most important mechanical state function of our model system is the energy, which is the sum of the kinetic energy and the potential energy ... 

The quantity p2 as a function of the coordinates is interpreted as the probability of the corresponding microscopic state of the system in this case the probability that the electron occupies a certain position relative to the nucleus. It is seen from equation 6 that in the normal state the hydrogen atom is spherically symmetrical, for p1M is a function of r alone. The atom is furthermore not bounded, but extends to infinity the major portion is, however, within a radius of about 2a0 or lA. In figure 3 are represented the eigenfunction pm, the average electron density p = p]m and the radial electron distribution D = 4ir r p for the normal state of the hydrogen atom. 

The density of states is the central function in statistical thermodynamics, and provides the key link between the microscopic states of a system and its macroscopic, observable properties. In systems with continuous degrees of freedom, the correct treatment of this function is not as straightforward as in lattice systems - we, therefore, present a brief discussion of its subtleties later. The section closes with a short description of the microcanonical MC simulation method, which demonstrates the properties of continuum density of states functions. 

To a chemist the entropy of a system is a macroscopic state function, i.e., a function of the thermodynamic variables of the system. In statistical mechanics, entropy is a mesoscopic quantity, i.e., a functional of the probability distribution, viz., the functional given by (V.5.6) and (V.5.7). It is never a microscopic quantity, because on the microscopic level there is no irreversibility. ... 

An isolated microscopic system is fully determined, in the quantum mechanical sense, when its state function y/ is known. In the Dirac formalism of quantum mechanics, the state function can be identified with a vector of state, 11//). (11) The system in the 1y/ state may equivalently be described by a Hermitian operator, the so-called density matrix p of a pure state,... 

Entropy is interpreted as the number of microscopic arrangements included in the macroscopic definition of a system. The second law is then used to derive the distribution of molecules and systems over their states. This allows macroscopic state functions to be calculated from microscopic states by statistical methods. 

We note that the classical equilibrium entropy (i.e., the eta-function evaluated at equilibrium states) acquires in the context of the Microcanonical Ensemble an interesting physical interpretation. The entropy becomes a logarithm of the volume of the phase space that is available to macroscopic systems having the fixed volume, fixed number of particles and fixed energy. If there is only one microscopic state that corresponds to a given macroscopic state, we can put the available phase space volume equal to one and the entropy becomes thus zero. The one-to-one relation between microscopic and macroscopic thermodynamic equilibrium states is thus realized only at zero temperature. 

This expression of the law of the realization of the microscopic states can be separated in two parts the partition function Zo of all species Ej(mj) and Ei(Mj) of which numbers are far above hundreds and the partition function Z(T,V,N) for species whose number is less than 100. The probability P(N) is then given by ... 

Thermodynamic properties or coordinates are derived from the statistical averaging of the observable microscopic coordinates of motion. If a thermodynamic property is a state function, its change is independent of the path between the initial and final states, and depends on only the properties of the initial and final states of the system. The infinitesimal change of a state function is an exact differential. 

Example 1.3 Entropy and distribution of probability Entropy is a state function. Its foundation is macroscopic and directly related to macroscopic changes. Such changes are mostly irreversible and time asymmetric. Contrary to this, the laws of classical and quantum mechanics are time symmetric, so that a change between states 1 and 2 is reversible. On the other hand, macroscopic and microscopic changes are related in a way that, for example, an irreversible change of heat flow is a direct consequence of the collision of particles that is described by the laws of mechanics. Boltzmann showed that the entropy of a macroscopic state is proportional to the number of configurations fl of microscopic states a system can have... 

Note that interpretations of the time-reversal experiments are only valid in strictly euclidean space-time. This condition is rarely emphasized by authors who state that all laws of physics are time-reversible, except for the law of entropy. Fact is that entropy is the only macroscopic state function which is routinely observed to be irreversible. One common explanation is to hint that entropy is an emergent property of macro systems and hence undefined for microsystems. Even so, the mystery of the microscopic origin of entropy remains. A plausible explanation may be provided if the assumed euclidean geometry of space-time is recognized as an approximate symmetry as demanded by general relativity. 

Density matrices of the state functions provide a compact graphical representation of important microscopic features for second order nonlinear optical processes. The transition moment y is expressed in terms of the transition density matrix p jji(r,r ) by nn " /j, ptr Pjj t(r,r )dr and the dipole moment difference Ay by the difference density function p - p between the excited and ground state functions = -e / r( p -p )dr where p is the first order reduced density matrix. 

Consider an equilibrium thennodynamic ensemble, say a set of atomic systems characterized by the macroscopic variables T (temperature), Q (volume), andTV (number of particles). Each system in this ensemble contains N atoms whose positions and momenta are assigned according to the distribution function (5.2) subjected to the volume restriction. At some given time each system in this ensemble is in a particular microscopic state that coiTesponds to a point (r, p- ) in phase space. As the system evolves in time such a point moves according to the Newton equations of motion, defining what we call a phase space trajectory (see Section 1.2.2). The ensemble coiTesponds to a set of such trajectories, defined by their starting point and by the Newton equations. Due to the uniqueness of solutions of the Newton s equations, these trajectories do not intersect with themselves or with each other. 

Since the goal of a simulation is to calculate properties of a particular macroscopic thermodynamic system from the configurations of many microscopic states, the final consideration in model selection is the choice of the thermodynamic state of the system. This can be done by fixing three thermodynamic variables (such as pressure, P, temperature T, internal energy, E) and designing the simulation so that these functions remain constant throughout the calculation. The choice of the thermodynamic state defines the ensemble and since the ensemble is chosen based upon the properties of interest, more details about the different ensembles will be given in Section 4.2.4.3. 

An exact knowledge of the microcanonical entropy, or the density of the states, of a protein model in both the native and nonnative states is crucial for a precise characterization of the folding process of the model, such as whether the folding is first-order or gradual, whether the model can fold uniquely to the native structure, whether there is a discontinuity in the order parameter of the conformation in the folding transition, and so on. Once the microscopic entropy function is accurately determined, the statistical mechanics of the protein folding problem is solved. The accurate determination of the entropy function of protein models by the ESMC method requires a proper treatment of the computational problems discussed in Section IV. 

Here qrot,s and qvib,s are the microscopic partition functions for rotation and vibration of M in the solution while Hm.s and Am, indicate the related numeral densities and the momentum partition functions, respectively. The functional (G ) has now the meaning of the free energy of the entire solute-solvent system, at the temperature T, with respect to a reference state given by non-interaction nuclei and electron, supplemented by the imperturbed pure liquid S, at the same temperature. Then, the fundamental energetic quantity connected with the insertion of the solute in the solvent, i.e. the free energy of solvation can be obtained as... 

The transition probability density j,j (x) is the only microscopically derived function that we need for milestoning. Note that we assume that is independent of the absolute time. This assumption is not valid in systems that strongly deviate from equilibrium or from a stationary state. Recently Vanden Eijnden et al. have shown that milestoning can be made mathematically exact if the microscopic dynamics is Brownian and the hypersurfaces are committers [16]. 

Explicitly, we look at some experiments seeking (a) to come fairly directly at the nonmechanical state functions of thermodynamics, such as the entropy and free energy, (b) to obtain information about quantum-mechanical fluids, and (c) to choose boundary conditions so that one obtains information not about the interior of a macroscopic system, but about some microscopic region. These three excursions are made in Sections 2, 3, and 4, respectively. Rather more attention is given to the first of the three topics than the others, since there has been more activity along this line. 

The microscopic state of a system may be specified in terms of the positions and momenta of the set of particles (atoms or molecules). Making the approximation that a classical description is adequate, the Hamiltonian H of a system of N particles can be written as a sum of a kinetic K and a potential V energy functions of the set of coordinates q and momenta Pi of each particle i, i.e. ... 

The previous summary provides the basic relationships, derived from the first and second laws, used for the manipulation of available experimental data. However, statistical thermodynamics is then required to develop expressions for the thermodynamic properties in terms of the fluctuating quantities of interest here. First, we will use statistical thermodynamics to provide the characteristic thermodynamic potentials in terms of the appropriate partition function, which will involve a sum over the microscopic states available to the systan. Second, we will provide relevant expressions for the fluctuations nnder one set of variables, which can then be used to rationalize the thermodynamic properties of a system characterized by a different set of variables. 

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