System Planck

Integral or differential entropy In classical thermodynamics, the concept of temperature is introduced first, then the concept of entropy, which is defined as the difference between two states of a system (Planck 1922). Most often, a sequence of transformations of an ideal system (perfect gas), called... 

In the previous section we discussed light and matter at equilibrium in a two-level quantum system. For the remainder of this section we will be interested in light and matter which are not at equilibrium. In particular, laser light is completely different from the thennal radiation described at the end of the previous section. In the first place, only one, or a small number of states of the field are occupied, in contrast with the Planck distribution of occupation numbers in thennal radiation. Second, the field state can have a precise phase-, in thennal radiation this phase is assumed to be random. If multiple field states are occupied in a laser they can have a precise phase relationship, something which is achieved in lasers by a teclmique called mode-locking Multiple frequencies with a precise phase relation give rise to laser pulses in time. Nanosecond experiments... 

Figure A2.2.3. Planck spectral density fimction as a fimction of the dimensionless frequency /)oi/(/rj 7). A2.2.4.7 APPLICATION TO IDEAL SYSTEMS ELASTIC WAVES IN A SOLID...

A linear dependence approximately describes the results in a range of extraction times between 1 ps and 50 ps, and this extrapolates to a value of Ws not far from that observed for the 100 ps extractions. However, for the simulations with extraction times, tg > 50 ps, the work decreases more rapidly with l/tg, which indicates that the 100 ps extractions still have a significant frictional contribution. As additional evidence for this, we cite the statistical error in the set of extractions from different starting points (Fig. 2). As was shown by one of us in the context of free energy calculations[12], and more recently again by others specifically for the extraction process [1], the statistical error in the work and the frictional component of the work, Wp are related. For a simple system obeying the Fokker-Planck equation, both friction and mean square deviation are proportional to the rate, and... 

These three terms represent contributions to the flux from migration, diffusion, and convection, respectively. The bulk fluid velocity is determined from the equations of motion. Equation 25, with the convection term neglected, is frequently referred to as the Nemst-Planck equation. In systems containing charged species, ions experience a force from the electric field. This effect is called migration. The charge number of the ion is Eis Faraday s constant, is the ionic mobiUty, and O is the electric potential. The ionic mobiUty and the diffusion coefficient are related ... 

In ion-exchange resins, diffusion is further complicated by electrical coupling effec ts. In a system with M counterions, diffusion rates are described by the Nernst-Planck equations (Helfferich, gen. refs.). Assuming complete Donnan exclusion, these equations canbe written... 

Parts of the calculations have been performed on the Cray systems of the Centro Interdipartimentale di Calcolo dell Universita di Trieste and of CINECA at Bologna. This work was supported by the Deutsche Forschungsgemeinschaft (DFG). The authors are also indebted to Prof. P. Fulde of the Max-Planck-Institut fiir komplexe Systeme, Dresden for support. 

Finally, we shall almost always use natural units in which c, the velocity of light, mid ft Planck s constant divided by 2it are set equal to one. In this system of units, energy, mass, inverse length, and inverse time all have the same dimension. 

We are very often concerned with magnitudes such as pressure, density, concentration, temperature, etc., which have the significance of mean values, and it must be remembered that wre cannot apply these terms to systems which are so constituted as to prohibit the existence of such a mean value. This point is by no means merely a logical or mathematical refinement, but is of the very essence of the physical interpretation of the second law of thermodynamics (cf. Planck, be. cit.). 

Criterion (1) is seen to be identical with Horstmann s principle it has been largely employed in the treatment of equilibria by Planck. It is, however, not always convenient in application because the systems which actually occur in practice are not isolated we shall therefore modify the relation so as to make it suitable for non-isolated systems. In this investigation we shall recover the first general method for determining the conditions of equilibrium—the principle of dissipation of energy. 

It may be remarked that there is no call for an atomic theory of energy, analogous to the atomic theories of matter and electricity, as the discontinuity arises from the peculiar character of the system (cf. Planck, 45, 5, 1912). 

Like the engine-based statements, Caratheodory s statement invokes limitations. From a given thermodynamic state of the system, there are states that cannot be reached from the initial state by way of any adiabatic process. We will show that this statement is consistent with the Kelvin-Planck statement of the Second Law. 

We wish to show that no points to the leftbb of 2 on the isotherm 62 are accessible from point 1 via any adiabatic path, reversible or irreversible. Suppose we assume that some adiabatic path does exist between 1 and 2. We represent this path as a dotted curve in Figure 2.11a. We then consider the cycle I —>2 —> 1 — 1. The net heat associated with this cycle would be that arising from the last step 1 — 1, since the other two steps are defined to be adiabatic. We have defined the direction 1 — 1 to correspond to an absorption of heat, which we will call qy. From the first law, the net work vv done in the cycle, is given by w = —q, since AU for the cycle is zero. Thus, for this process, iv is negative (and therefore performed by the system), since qy is positive, having been absorbed from the reservoir. The net effect of this cycle, then, is to completely convert heat absorbed at a high temperature reservoir into work. This is a phenomenon forbidden by the Kelvin-Planck statement of the Second Law. Hence, points to the left of 2 cannot be reached from point 1 by way of any adiabatic path. 

ESO VLT/Max Planck CW Dye Laser. The MPI is developing a CW dye laser for deployment on one ESO 8-m VLT telescope in 2004 (Eig. 13). The oscillator is a Coherent 899 ring dye laser, with a 2-5 W output, pumped by a 10 W, Coherent Verdi frequency-doubled Nd YAG laser. The beam is amphfied in a four-pass amphfier with 4 high velocity dye jets pumped with 4 10 W Verdi lasers. The system utihzes Rhodamine 6G in ethylene glycol however, because of the high pump power, the dye degrades quickly, and must... 

Using Bohr s model, one could calculate the energy difference between orbits of an electron in a hydrogen atom with Planck s equation. In the example of a system with only two possible orbits, the equation of the emitted radiation as the electron went from a higher energy state 2 to a lower one j would be - E = hf, where h is Planck s constant and/is the frequency of the emitted radiation. 

A useful expression for evaluating expectation values is known as the Hell-mann-Feynman theorem. This theorem is based on the observation that the Hamiltonian operator for a system depends on at least one parameter X, which can be considered for mathematical purposes to be a continuous variable. For example, depending on the particular system, this parameter X may be the mass of an electron or a nucleus, the electronic charge, the nuclear charge parameter Z, a constant in the potential energy, a quantum number, or even Planck s constant. The eigenfunctions and eigenvalues of H X) also depend on this... 

All equations given in this text appear in a very compact form, without any fundamental physical constants. We achieve this by employing the so-called system of atomic units, which is particularly adapted for working with atoms and molecules. In this system, physical quantities are expressed as multiples of fundamental constants and, if necessary, as combinations of such constants. The mass of an electron, me, the modulus of its charge, lei, Planck s constant h divided by lit, h, and 4jt 0, the permittivity of the vacuum, are all set to unity. Mass, charge, action etc. are then expressed as multiples of these constants, which can therefore be dropped from all equations. The definitions of atomic units used in this book and their relations to the corresponding SI units are summarized in Table 1-1. 

---