Constants of motion

A statistical ensemble can be viewed as a description of how an experiment is repeated. In order to describe a macroscopic system in equilibrium, its thennodynamic state needs to be specified first. From this, one can infer the macroscopic constraints on the system, i.e. which macroscopic (thennodynamic) quantities are held fixed. One can also deduce, from this, what are the corresponding microscopic variables which will be constants of motion. A macroscopic system held in a specific thennodynamic equilibrium state is typically consistent with a very large number (classically infinite) of microstates. Each of the repeated experimental measurements on such a system, under ideal... 

A stationary ensemble density distribution is constrained to be a functional of the constants of motion (globally conserved quantities). In particular, a simple choice is pip, q) = p (W (p, q)), where p (W) is some fiinctional (fiinction of a fiinction) of W. Any such fiinctional has a vanishing Poisson bracket (or a connnutator) with Wand is thus a stationary distribution. Its dependence on (p, q) through Hip, q) = E is expected to be reasonably smooth. Quanttun mechanically, p (W) is die density operator which has some fiinctional dependence on the Hamiltonian Wdepending on the ensemble. It is also nonnalized Trp = 1. The density matrix is the matrix representation of the density operator in some chosen representation of a complete orthononnal set of states. If the complete orthononnal set of eigenstates of die Hamiltonian is known ... 

The microcanonical ensemble is a certain model for the repetition of experiments in every repetition, the system has exactly the same energy, Wand F but otherwise there is no experimental control over its microstate. Because the microcanonical ensemble distribution depends only on the total energy, which is a constant of motion, it is time independent and mean values calculated with it are also time independent. This is as it should be for an equilibrium system. Besides the ensemble average value (il), another coimnonly used average is the most probable value, which is the value of tS(p, q) that is possessed by the largest number of systems in the ensemble. The ensemble average and the most probable value are nearly equal if the mean square fluctuation is small, i.e. if... 

Fluctuations of observables from their average values, unless the observables are constants of motion, are especially important, since they are related to the response fiinctions of the system. For example, the constant volume specific heat of a fluid is a response function related to the fluctuations in the energy of a system at constant N, V and T, where A is the number of particles in a volume V at temperature T. Similarly, fluctuations in the number density (p = N/V) of an open system at constant p, V and T, where p is the chemical potential, are related to the isothemial compressibility iCp which is another response fiinction. Temperature-dependent fluctuations characterize the dynamic equilibrium of themiodynamic systems, in contrast to the equilibrium of purely mechanical bodies in which fluctuations are absent. 

In addition to affecting the number of active degrees of freedom, the fixed n also affects the iinimolecular tln-eshold E in). Since the total angular momentum j is a constant of motion and quantized according to... 

Shirts R B and Reinhardt W P 1982 Approximate constants of motion for classically chaotic vibrational dynamics vague tori, semiclassical quantization, and classical intramolecular energy flow J. Cham. Phys. 77 5204-17... 

For long-term simulations, it generally proves advantageous to consider numerical integrators which pass the structural properties of the model onto the calculated solutions. Hence, a careful analysis of the conservation properties of QCMD model is required. A particularly relevant constant of motion of the QCMD model is the total energy of the system... 

Conservation of energy. Assuming that U and H do not depend explicitly on time or velocity (so that dH/dt = 0), it is easy to show from Eq. (8) that the total derivative dFUdt is zero i.e., the Hamiltonian is a constant of motion for Newton s equation. In other words, there is conservation of total energy under Newton s equation of motion. 

The best gauges for the stability of any simulation are the constants of motion of the physical equation that is numerically solved, i.e., quantities that are expected to be conserved during the simulation. Since numerical fluctuations cannot be avoided, a dy-... 

Conservation of linear and angular momenta. After equilibrium is reached, the total linear momentum P [Eq. (9)] and total angular momentum L [Eq. (10)] also become constants of motion for Newton s equation and should be conserved. In advanced simulation schemes, where velocities are constantly manipulated, momentum conservation can no longer be used for gauging the stability of the simulation. 

We have exhibited two constants of the motion that can be diagonalized simultaneously with h. A third constant of motion is the parity operator... 

Angular momentum plays an important role in both classical and quantum mechanics. In isolated classical systems the total angular momentum is a constant of motion. In quantum systems the angular momentum is important in studies of atomic, molecular, and nuclear structure and spectra and in studies of spin in elementary particles and in magnetism. 

In order for an ensemble to represent a system in equilibrium, the density of phase must remain constant in time. This means that Liouville s equation is satisfied, which requires that g is constant with respect to the coordinates q and p, or that g = g(a), where a = a(q,p) is a constant of the motion. Constants of motion are properties of an isolated system which do not change with time. Familiar examples are energy, linear momentum and angular momentum. For constants of motion H,a = 0. Hence, if g = g a) and a is a constant of motion, then... 

By letting AE —> 0 one gets a surface ensemble in which all systems have exactly the energy E. g(E) is in statistical equilibrium because E is a constant of motion. 

The potential energy functions are expressed in terms of q, 0 = 1,2,. 6), which explicitly exhibits its independence of the coordinates of the center of mass. Again, since the momenta conjugate to coordinates 0/7. q, qg), i.e. Pi,p andp9 remains constants of motion during the entire collision, the term containing them in the Hamiltonian has been subtracted. 

In addition to the study of atomic motion during chemical reactions, the molecular dynamics technique has been widely used to study the classical statistical mechanics of well-defined systems. Within this application considerable progress has been made in introducing constraints into the equations of motion so that a variety of ensembles may be studied. For example, classical equations of motion generate constant energy trajectories. By adding additional terms to the forces which arise from properties of the system such as the pressure and temperature, other constants of motion have been introduced. 

In the Schwinger representation the identity operator in the spin Hilbert space is mapped onto the constant of motion a a + a a2. The existence of this constant of motion is utilized by the Holstein-Primakoff transformation to eliminate one boson DoF, thus representing the spin DoF by a single oscillator [97] ... 

On the negative side, the exact time dependent centroid Hamiltonian in Eq. (44) is a constant of motion and the CMD method does not satisfy this condition in general except for quadratic potentials. 

In the simplest case of a free evolution without damping, pumping, and mismatch, the equations of motion (3) are solved analytically. One easily notes that the system (3) now belongs to the class of Hamiltonian systems with two constants of motion ... 

The system (13)—(14) has two independent constants of motion (first integrals) the Hamilton function (10) and... 

It can be easily checked by direct computation that we have really obtained a realization of the Lie algebra g in a Hilbert (Fock) space, [T a, T fc] = ifabc fc, in accordance with (11), where Ta = T f/aL . For an irreducible representation R, the second-order Casimir operator C2 is proportional to the identity operator I, which, in turn, is equal to the number operator N in our Fock representation, that is, if T" —> Ta, then I /V 5/,/a . Thus we obtain an important for our further considerations constant of motion N ... 

Since the direction of the interaction force, -VF, is along the relative position vector, R, the vector product of R with R, Eq. 5.68, vanishes, RxR = 0. In other words, RxmR = Rxmv = L, the angular momentum of relative motion, is a constant of motion. Its magnitude may be expressed in polar coordinates R, 3, according to... 

This point was forcefully argued by P. and T. Ehrenfest in their famous article quoted in V.4. Yet it is still often ignored and then leads to the paradox that the entropy is a constant of motion. 

The finer structure within each feature state corresponds to the dynamics of the Franck-Condon bright state within a four-dimensional state space. This dynamics in state space is controlled by the set of all known anharmonic resonances. The state space is four dimensional because, of the seven vibrational degrees of freedom of a linear four-atom molecule, three are described by approximately conserved constants of motion (the polyad quantum numbers) thus 7-3 = 4. 

What is a polyad A polyad is a subset of the zero-order states within a specifiable region of Evib (typically a few hundred reciprocal centimeters) that are strongly coupled by anharmonic resonances to each other and negligibly coupled to all other nearby zero-order states. If approximate constants of motion of the exact vibration-rotation Hamiltonian exist, then the exact H can be (approximately) block diagonalized. Each subblock of H corresponds to one polyad and is labeled by a set of polyad quantum numbers. For the C2H2S0 state, a procedure proposed by Kellman [9, 10] identifies the three polyad quantum numbers... 

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