Forces, external

In Chapter III, surface free energy and surface stress were treated as equivalent, and both were discussed in terms of the energy to form unit additional surface. It is now desirable to consider an independent, more mechanical definition of surface stress. If a surface is cut by a plane normal to it, then, in order that the atoms on either side of the cut remain in equilibrium, it will be necessary to apply some external force to them. The total such force per unit length is the surface stress, and half the sum of the two surface stresses along mutually perpendicular cuts is equal to the surface tension. (Similarly, one-third of the sum of the three principal stresses in the body of a liquid is equal to its hydrostatic pressure.) In the case of a liquid or isotropic solid the two surface stresses are equal, but for a nonisotropic solid or crystal, this will not be true. In such a case the partial surface stresses or stretching tensions may be denoted as Ti and T2-... 

Another possibility is that a system may be held in a constrained equilibrium by external forces and thus be in a non-equilibrium steady state (NESS). In this case, the spatio-temporal correlations contain new ingredients, which are also exemplified in section A3.3.2. 

The fluctuation dissipation theorem relates the dissipative part of the response fiinction (x") to the correlation of fluctuations (A, for any system in themial equilibrium. The left-hand side describes the dissipative behaviour of a many-body system all or part of the work done by the external forces is irreversibly distributed mto the infinitely many degrees of freedom of the themial system. The correlation fiinction on the right-hand side describes the maimer m which a fluctuation arising spontaneously in a system in themial equilibrium, even in the absence of external forces, may dissipate in time. In the classical limit, the fluctuation dissipation theorem becomes / /., w) = w). 

The tliree conservation laws of mass, momentum and energy play a central role in the hydrodynamic description. For a one-component system, these are the only hydrodynamic variables. The mass density has an interesting feature in the associated continuity equation the mass current (flux) is the momentum density and thus itself is conserved, in the absence of external forces. The mass density p(r,0 satisfies a continuity equation which can be expressed in the fonn (see, for example, the book on fluid mechanics by Landau and Lifshitz, cited in the Furtlier Reading)... 

Smoluchowski theory [29, 30] and its modifications fonu the basis of most approaches used to interpret bimolecular rate constants obtained from chemical kinetics experiments in tenus of difhision effects [31]. The Smoluchowski model is based on Brownian motion theory underlying the phenomenological difhision equation in the absence of external forces. In the standard picture, one considers a dilute fluid solution of reactants A and B with [A] [B] and asks for the time evolution of [B] in the vicinity of A, i.e. of the density distribution p(r,t) = [B](rl)/[B] 2i ] r(t))l ] Q ([B] is assumed not to change appreciably during the reaction). The initial distribution and the outer and inner boundary conditions are chosen, respectively, as... 

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

As witli tlie nematic phase, a chiral version of tlie smectic C phase has been observed and is denoted SniC. In tliis phase, tlie director rotates around tlie cone generated by tlie tilt angle [9,32]. This phase is helielectric, i.e. tlie spontaneous polarization induced by dipolar ordering (transverse to tlie molecular long axis) rotates around a helix. However, if tlie helix is unwound by external forces such as surface interactions, or electric fields or by compensating tlie pitch in a mixture, so tliat it becomes infinite, tlie phase becomes ferroelectric. This is tlie basis of ferroelectric liquid crystal displays (section C2.2.4.4). If tliere is an alternation in polarization direction between layers tlie phase can be ferrielectric or antiferroelectric. A smectic A phase foniied by chiral molecules is sometimes denoted SiiiA, altliough, due to the untilted symmetry of tlie phase, it is not itself chiral. This notation is strictly incorrect because tlie asterisk should be used to indicate the chirality of tlie phase and not tliat of tlie constituent molecules. 

Other external forces or potentials can also be used, e.g., constant forces, or torques applied to parts of a protein to induce rotational motion of its domains (Wriggers and Schulten, 1997a). 

The simulation (Lu et al., 1998) suggested how Ig domains achieve their chief design requirement of bursting one by one when subjected to external forces. At small extensions, the hydrogen bonds between strands A and B and between strands A and G prevent significant extension of a domain, i.e.. 

In this section we describe the behavior of a ligand subjected to three types of external forces a constant force, forces exerted by a moving stiff harmonic spring, and forces exerted by a soft harmonic spring. We then present a method of reconstruction of the potential of mean force from SMD force measurements employing a stiff spring (Izrailev et al., 1997 Balsera ct al., 1997). 

In the limit of very slow change (quasi-static process) the frictional component is zero and then the work done by the external force equals the free energy change, i.e.. 

In physical systems, in the absence of external forces, A and B approach equilibrium ... 

As our first model problem, we take the motion of a diatomic molecule under an external force field. For simplicity, it is assumed that (i) the motion is pla nar, (ii) the two atoms have equal mass m = 1, and (iii) the chemical bond is modeled by a stiff harmonic spring with equilibrium length ro = 1. Denoting the positions of the two atoms hy e 71, i = 1,2, the corresponding Hamiltonian function is of type... 

It is assumed that a> and v remain unchanged at time step since there is no external forces. 

The first chapter, on Conformational Dynamics, includes discussion of several rather recent computational approaches to treat the dominant slow modes of molecular dynamical systems. In the first paper, SCHULTEN and his group review the new field of steered molecular dynamics (SMD), in which large external forces are applied in order to be able to study unbinding of ligands and conformation changes on time scales accessible to MD... 

The starting point for developing the model is the set of diffusion equations for a gas mixture in the presence of temperature, pressure and composition gradients, and under the influence of external forces." These take the following form... 

In order to satisfy requirement (b) the dust molecules must be constrained by external forces Pliysically these represent the forces exerted... 

Now the force per unit volume exerted on the porous medium by the pressure gradient in the gas is -grad p, where p, as distinct from is the physical pressure of the gaseous mixture. This is the force which must be balanced in our model by the external forces acting on the dust particles, so... 

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