Ehrenfest definition

The transitions between the bottom five phases of Fig. 2 may occur close to equilibrium and can be described as thermodynamic first order transitions (Ehrenfest definition 17)). The transitions to and from the glassy states are limited to the corresponding pairs of mobile and solid phases. In a given time frame, they approach a second order transition (no heat or entropy of transition, but a jump in heat capacity, see Fig. 1). 

We shall use the principle of stationary action to obtain a variational definition of the force acting on an atom in a molecule. This derivation will illustrate the important point that the definition of an atomic property follows directly from the atomic statement of stationary action. To obtain Ehrenfest s second relationship as given in eqn (5.24) for the general time-dependent case, the operator G in eqn (6.3) and hence in eqn (6.2) is set equal to pi, the momentum operator of the electron whose coordinates are integrated over the basin of the subsystem 1. The Hamiltonian in the commutator is taken to be the many-electroii, fixed-nucleus Hamiltonian... 

The atomic statements of the Ehrenfest force law and of the virial theorem establish the mechanics of an atom in a molecule. As was stressed in the derivations of these statements, the mode of integration used to obtain an atomic average of an observable is determined by the definition of the subsystem energy functional i2]. It is important to demonstrate that the definition of this functional is not arbitrary, but is determined by the requirement that the definition of an open system, as obtained from the principle of stationary action, be stated in terms of a physical property of the total system. This requirement imposes a single-particle basis on the definition of an atom, as expressed in the boundary condition of zero flux in the gradient vector field of the charge density, and on the definition of its average properties. 

Ehrenfest s first relation, the definition of the velocity and its associated momentum, is obtained using the commutator of H and the position vector r. This commutator, using the above Hamiltonian and multiplied by m, does indeed yield k as the momentum of an electron in an electromagnetic field... 

The virial theorem plays a dominant role in the definition of pressure, in both classical and quantum mechanics. The following section demonstrates that the pv product for a proper open system is proportional to the surface virial, Equation (9), the virial of the Ehrenfest forces exerted by the surroundings on the open system [9,12],... 

The total virial for an open system v(Q), is of course, origin independent since it equals —27 ( 2). This is not the case, however, for its expression in terms of the sum of its basin v ( 2) and surface v,v( 2) contributions, whose values, according to their definitions in Equations (8) and (9), are dependent up a choice of origin. Consider a coordinate transformation denoted by r = r + 5R caused by a shift SR in the origin. This has the effect of changing the basin and surface virials by the same absolute amount, equal to the virial of the Ehrenfest force as given in Equation (23),... 

While Pendas makes no attempt to disprove the quantum definition of pressure obtained through the scaling procedure of Marc and McMillan [22] as presented here, he does state that "the use of electron-only scaling to study stressed situations, where the virials due to the nuclear system cannot be neglected, is not a very consistent procedure." In reality, the physics of an open system does not neglect the virials due to the nuclear system, they are included in the total virial that is defined by taking the virial of the Ehrenfest force, see for example Equation (13). The virial theorem illustrates... 

The first approximation made in the Ehrenfest method is thus the factorisation of the total wavefunction into a product of electronic and nuclear parts. One deficiency of the ansatz (2) is the fact that the electronic wavefunction does not have the possibility to decohere the populated electronic states in P(r,t) share the same nuclear wave-packet x(R, t) by definition of the total wavefunction. Decoherence here is defined as the tendency of the time-evolved electronic wavefunction to behave as a statistical ensemble of electronic states rather than a coherent superposition of them [26]. The neglect of electronic decoherence could lead to non-physical asymptotic behaviors in case of bifurcating paths. It is not expected to be a problem here as we are interested in relatively short timescale dynamics. 

Dimarzio [18] identified this state of vanishing entropy as the entropy catastrophe introduced by Kauzmann [19], and regarded T2 as the glass transition temperature Tg of the polymer. Because the temperature derivative of the entropy is discontinuous if the entropy is kept constant at A5" = 0 below T2, the glass transition on the basis of this picture is classified into a second-order transition by Ehrenfest s definition. 

To conclude, we have seen that for a given wave function and Hamiltonian, the Ehrenfest theorem can be instrumentalized to derive explicit expressions for the density and current-density distributions by rewriting it in such a way that the continuity equation results. We will rely on this option in the relativistic framework in chapters 5, 8, and 12 to define these distributions for relativistic Hamiltonian operators and various approximations of N-particle wave functions. From the derivation, it is obvious that the definition of the current density is determined by the commutator of the Hamiltonian operator with the position operator of a particle. All terms of the Hamiltonian which depend on the momentum operator of the same particle will produce contributions to the current density. In section 5.4.3 we will encounter a case in which the momentum operator is associated with an external vector potential so that an additional term will show up in the commutator. Then, the definition of the current density has to be extended and the additional term can be attributed to an (external-field) induced current density. 

Sometimes it is stiU debated whether the glass transition is a purely kinetic transition or a second-order thermodynamic transition (van Krevelen 2003). On one hand, it is true that the crystallization process for a number of (atactic) polymers would not take place even at infinite time, and this transition possesses the characteristics of a second-order thermodynamic transition (at least formally, in the Ehrenfest sense see definition of the phase transition in Section 2.2). But the absence of crystallization does not prove that the glass transition is a thermodynamic second-order transition, and it is also true that the glass transition does not occur as a definite sharp transition as would be required by equilibrium thermodynamics. Therefore, the glass transition must be considered a kinetic transition. 

According to the definition of adiabaticity given by Ehrenfest (Pake, 1962), adiabatic passage through the resonance will occur when... 

Pendas AM, Hemandez-Trujillo J (2012) The Ehrenfest force field topology and consequences for the definition of an atom in a molecule. J Chem Phys 137 134101... 

Any definition of a (r) whose divergence gives the correct Ehrenfest force is acceptable. That is, the stress tensor is an inherently ambiguous quantity. In particular, every possible divergence-free tensor, V G (r) = 0, provides an alternative definition of the stress tensor, namely ... 

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