Boltzman distributions

Consider a physical system with a set of states a, each of which has an energy Hio). If the system is at some finite temperature T, random thermal fluctuations will cause a and therefore H a) to vary. While a system might initially be driven towards one direction (decreasing H, for example) during some transient period immediately following its preparation, as time increases, it eventually fluctuates around a constant average value. When a system has reached this state, it is said to be in thermal equilibrium. A fundamental principle from thermodynamics states that when a system is in thermal equilibrium, each of its states a occurs with a probability equal to the Boltzman distribution P(a)  

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Measurement of the energy difference is achieved by a resonance method. The population of nuclei in a given state is governed by the Boltzman distribution that leg s to an of nuclei in the state of lowest energy and... 

So far we have discussed mainly stable configurations that have reached an equilibrium. What about the evolution of a system from an arbitrary initial state In particular, what do we need to know in order to be assured of reaching an equilibrium state that is described by the Boltzman distribution (equation 7.1) from an arbitrary initial state It turns out that it is not enough to know just the energies H ct) of the different states a. We also need to know the set of transition probabilities between ail pairs of states of the system. 

A gas is not in equilibrium when its distribution function differs from the Maxwell-Boltzman distribution. On the other hand, it can also be shown that if a system possesses a slight spatial nonuniformity and is not in equilibrium, then the distribution function will monotonically relax in velocity space to a local Maxwell-Boltzman distribution, or to a distribution where p = N/V, v and temperature T all show a spatial dependence [bal75]. 

In order to get this expression into a more familiar form (equation 9.7), we now consider the zeroth-order approximation to /. We assume that / is locally a Maxwell-Boltzman distribution, and treat the density p, temperature T[x,t) = < V — u p> (where k is Boltzman s constant), and average velocity u all as slowly changing variables with respect to x and t. We can then write... 

Using the fact that we have a well defined energy function (equation 10.9), we know from statistical mechanics that when the system has reached equilibrium, the probability that it is in some state S = Si, S, , Sm) is given by the Boltzman distribution ... 

This prescription, which follows directly from the Boltzman distribution, ensures that any update of the system for T > 0 has a finite probability of increasing the energy of the system and thereby of helping the system get out of local minima. The lower the temperature, the less likely are higher energy states as the temperature... 

Top typical saturation curve and variation of mean electron energy with applied field. Middle fraction of the electron swarm exceeding the specific energy at each field strength. Calculated assuming constant collision cross-section and Maxwell-Boltzman distribution. Bottom variation of products typical of involvement of ionic precursors (methane) and excited intermediates (ethane) with applied field strength... 

Using small datasets we initially showed that the dynamic PSA (PSA[Pg.347]

The Maxwell-Boltzman distribution function permits calculation of the effect of temperature on each electronic transition. Designated by /V0 and /Ve, the number of atoms in the ground and excited states, one obtains ... 

The apparatus has been already described in Ref. [7]. The clusters are produced by laser vaporization of a sodium rod, with helium at about 5 bars as a carrier gas and a small amount of SF6. The repetition rate is 10 Hz. In this configuration, the vibrationnal temperature of the formed clusters is roughly 400 K,[10] that gives 85% of C2V geometry and 15% of C3V for a Boltzman distribution. The laser beams are focused onto the cluster beam between the first two plates of an axial Wiley Mac-Laren Time-Of-Flight mass spectrometer with a reflectron. The photoionization efficiency curve as well as the photoabsorption spectrum determined by a photodepletion experiement are displayed on Fig. 1(b) and 1(c) respectively. The ionization threshold is at 4.3 eV, close to the 4.4 eV calculated for the C3V isomer and 4.9 eV for the C2V isomer (see the Fig. 1 (b)). The conclusion arising out of the photodepletion spectrum shown on Fig 1(c) and from ab initio calculations of the excited states, [5] is that the observed... 

The Gouy—Chapman treatment combines the Poisson equation in onedimensional form, which relates the electrical potential (x) to the charge density at x, and the Boltzman distribution of ions in thermal motion [6,18]... 

Replacing the Fermi—Dirac distribution by the Boltzman distribution (for U below the Fermi level) and integrating eqn. (151), a current maximum close to the Fermi level of the electrode is predicted with the major contribution within kB T around UF. 

This equation describes an equilibrium distribution of the electroinactive ion in an electrostatic field. This is a typical situation to which one may apply the Boltzman distribution law, which states... 

This difference between fermions and bosons is reflected in how they occupy a set of states, especially as a function of temperature. Consider the system shown in Figure E.10. At zero temperature (T = 0), the bosons will try to occupy the lowest energy state (a Bose-Einstein condensate) while for the fermions the occupancy will be one per quantum state. At high temperatures the distributions are similar and approach the Maxwell Boltzman distribution. 

Entropy is a measure of the degree of randomness in a system. The change in entropy occurring with a phase transition is defined as the change in the system s enthalpy divided by its temperature. This thermodynamic definition, however, does not correlate entropy with molecular structure. For an interpretation of entropy at the molecular level, a statistical definition is useful. Boltzmann (1896) defined entropy in terms of the number of mechanical states that the atoms (or molecules) in a system can achieve. He combined the thermodynamic expression for a change in entropy with the expression for the distribution of energies in a system (i.e., the Boltzman distribution function). The result for one mole is ... 

Logic would suggest that A t must be inversely proportional to the average velocity, V, of the atoms in the system. Following the Boltzman distribution function [14] the number of atoms dNv, with velocity, V (independent of direction and expressed in polar coordinates) can be written as ... 

To apply this concept for a simple diatomic molecule, let s consider the example given in Figure 3. At room temperature, according to the Boltzman distribution, most of the molecules are in the lowest vibrational level (v) of the ground state (i.e., v = 0). The absorption spectrum presented in Figure 3b exhibits, in addition to the pure electronic transition (the so-called 0-0... 

Fig. 23.—Illustration of the energy supplied from an outside source in photochemical reactions. The normal thermal energy supplied from collisions by the Maxwell-Boltzman distribution is insufficient to produce the chemical reaction.

Z Partition function for all active degrees of freedom of the molecule /(e) The energy distribution function for the system of interest K(e) The thermal Boltzman distribution function... 

Based on a Boltzman distribution an energy difference of 10 kJ/mole between two conformers already leads to an almost exclusive population of the less strained species. 

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