Energy conjugated variables

Obviously, the general procedure to get a capacity is to formulate a thermodynamic function initially in its natural variables, say (7(5, V, n). Then a variable is replaced by its energy-conjugated variable, such as U(T, V,n), and the partial differential with respect to this variable is formed to get the capacity, namely dU T, V, n)/dT. 

The entropy and the temperature are energy-conjugated variables, in the same way as the volume and the pressure. Thus, Eq. (6.67) resembles somehow the osmotic... 

We made here again use of the law of Schwarz. We can represent the process now in refined representation either in F—h diagram or in tx-n diagram, cf. Fig. 9.2. As the process of the paddle wheel can be represented with two different energy conjugated variables, also, for example, the Carnot process can be represented in... 

There is a second characteristic property which all cases have in common One always deals with a pair of energy conjugated variables, that is to say, the displacement caused by the field results in work. More specifically, if a field gives rise to a displacement dx, then the work per unit volume is... 

The mean-squared fluctuations in thermal equilibrium can be calculated as for any pair of energy conjugated variables, by applying Boltzmann statistics. The probability distribution for (j)k is... 

The energy q of a nuclear or electronic excited state of mean lifetime t cannot be determined exactly because of the limited time interval At available for the measurement. Instead, q can only be established with an inherent uncertainty, AE, which is given by the Heisenberg uncertainty relation in the form of the conjugate variables energy and time,... 

The use of the electron density and the number of electrons as a set of independent variables, in contrast to the canonical set, namely, the external potential and the number of electrons, is based on a series of papers by Nalewajski [21,22]. A.C. realized that this choice is problematic because one cannot change the number of electrons while the electron density remains constant. After several attempts, he found that the energy per particle possesses the convexity properties that are required by the Legendre transformations. When the Legendre transform was performed on the energy per particle, the shape function immediately appeared as the conjugate variable to the external potential, so that the electron density was split into two pieces that can be varied independent the number of electrons and their distribution in space. 

The fundamental equation for U is in agreement with the statement of the preceding section that for a homogeneous mixture of Ns substances, the state of the system can be specified by Ns + 2 properties, at least one of which is extensive. The total number of variables involved in equation 2.2-8 is 2NS + 5. Ns + 3 of these variables are extensive (U, S, V, and (nj), and Ns + 2 of the variables are intensive (T, P, /.q ). Note that except for the internal energy, these variables appear in pairs, in which one property is extensive and the other is intensive these are referred to as conjugate pairs. These pairs are given later in Table 2.1 in Section 2.7. When other kinds of work are involved, there are more than 2Ns + 5 variables in the fundamental equation for U (see Section 2.7). 

Energy = Intensive variable x Conjugate extensive variable. (2.1)... 

In conclusion, entropy is the physical quantity that represents the capacity of distribution of energy over the energy levels of the individual constituent particles in the system. The extensive variable entropy S and the intensive variable the absolute temperature Tare conjugated variables, whose product TdS represents the heat reversibly transferred into or out of the system. In other words, the reversible transfer of heat into or out of the system is always accompanied by the transfer of entropy. 

The observation of a quantum-mechanical system involves the disturbance of the state being observed the Heisenberg6 uncertainty principle [5] dictates that the uncertainty Ax in position x and the uncertainty Apx in momentum px in the x direction (or in y or in z, or the uncertainty in any two "canonically conjugate" variables, e.g. energy E and time f, or angular momentum L and phase (f>, i.e. variables whose... 

In Eqs. (19)-(21), always the sign has to be chosen which yields the absolute minimum of the free energy. As usual, mean field theory yields metastable branches, and the phase separation from one state 4>coex (D) to the other state I coex (D) in this treatment shows up as an intersection of two branches for AF/kBT when plotted vs Ap or the conjugate variable A=b- [Pg.14]

We first inquire whether the state corresponding to 7 = 0 can be reached. Consider a system characterized by a deformation coordinate z with a conjugate variable Z such that the element of work is given by dW = —Zdz. Then the energy of the system is expressed functionally by E = E(S, z) = E(S(T, z), z) thus. 

The thermodynamic potential of the canonical ensemble, the Helmholtz free energy, is the first thermodynamic potential g=F, which is a function of the variables of state u 1 = T, x2=V, x3=N, and x4=z. It is obtained from the fundamental thermodynamic potential / =E (the energy) by the Legendre transform (Eq. (7)), exchanging the variable of state x1 =S of the fundamental thermodynamic potential with its conjugate variable u 1 = / . In the canonical ensemble, the first partial derivatives (Eq. (1)) of the fundamental thermodynamic potential are defined asu2=-p, u3=p, and u 4 = - S. The entropy (Eq. (46)) for the Tsallis and Boltzmann-Gibbs statistics in the canonical ensemble can be rewritten as... 

Our goal is now to go from the potential as our basic variable, to a new variable, which will be the electron density. The deeper reason that this is possible is that the density and the potential are conjugate variables. With this we mean that the contribution of the external potential to the total energy is simply an integral of the potential times the density. We make use of this relation if we take the functional derivative of the energy functional [v] with respect to the potential v ... 

We summarize in Table 2.1 the physical dimensions of various energy forms. We emphasize that the list is not complete. In square brackets the physical dimension of the extensive variable has been given. The intensive variable has the energy-conjugated physical dimension. In the following sections, we discuss briefly some issues of the enagy forms detailed in Table 2.1. 

From the foregoing it is appealing to relate generalized susceptibilities xj as the differential of an arbitrary extensive variable X with respect to its energy-conjugated intensive variable... 

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