Time-dependent external forcing

Box 21.3 Time-Dependent External Forcing of Linear One-Box-Model Box 21.4 Temporal Variability of PCE-Input and Response of Concentration in Greifensee... 

Time-Dependent External Forcing of Linear One-Box Model... 

In linear response theory, it is assumed that a time dependent external force F t) couples to an observable A (self-adjoint operator) and the response of the system to linear order in the external force is computed. More specifically, the Hamiltonian in the presence of the external force is Hit) = H — AF t), and the evolution equation for the density matrix is... 

For a closed, isolated system H is time independent time dependence in the Hamiltonian enters via effect of time-dependent external forces. Here we focus on the earlier case. Equation (1) is a first-order linear differential equation that can be solved as an initial value problem. If (to) is known, a formal solution to Eq. (1) is given by... 

We begin by restricting ourselves to the special case where the potential energy V is not a function of time but depends only on x. This will be true if the system experiences no time-dependent external forces. The time-dependent Schrbdinger equation reads... 

If there is no time-dependent external force, the dynamics of a molecular system will evolve on a constant-energy surface. Therefore, a natural choice of the statistical ensemble in molecular... 

The most frequently encountered numerical problem in nonlinear chemical dynamics is that of solving a set of ordinary, nonlinear, first-order, coupled, autonomous differential equations, such as those describing the BZ reaction. We hope you understand by now what nonlinear means, but let us comment on the other modifiers. The equations are ordinary because they do not contain partial derivatives (we consider partial differential equations in the next section), first order because the highest derivative is the first derivative, and coupled because the time derivative of one species depends on the concentrations of other species. In the absence of time-dependent external forcing, rate equations are autonomous, meaning that time does not appear explicitly on the right-hand side. 

Let us consider a situation in which a time-dependent external force is applied to the GGS beads see the Langevin equation of motion, Eq. 2. Now, averaging this equation over the thermal noise components we have for Ri t)) ... 

If there is no time-dependent external force, the dynamics of a molecular system will evolve on a constant-energy surface. Therefore, a natural choice of the statistical ensemble in molecular dynamics simulation is the micro-canonical ensemble (NVE). Other types of ensembles, such as the canonical ensemble (NVT) and the isothermal-isobaric ensemble (NPT), can also be realized by controlling corresponding thermodynamic variables. For the last two ensembles, the temperature of the ensemble needs to be controlled and four different control mechanisms, namely differential control, proportional control, integral control and stochastic control, have been developed in the literature. As an example, a proportional thermostat for the NVT ensemble will be briefly discussed as follows. 

The non linear viscoelasticity of various particles filled rubber is addressed in range of studies. It is found that the carbon black filled-elastomer exhibit quasi-static and dynamic response of nonlinearity. Hartmann reported a state of stress which is the superposition of a time independent, long-term, response (hyperelastic) and a time dependent, short-term, response in carbon black filled-rubber when loaded with time-dependent external forces. The short term stresses were larger than the long term hyperelastic ones. The authors had done a comparative study for the non linear viscoelastic models undergoing relaxation, creep and hysteresis tests [20-22]. For reproducible and accurate viscoelastic parameters an experimental procedure is developed using an ad hoc nonlinear optimization algorithm. 

Indeed, carbon black-filled rubber, when loaded with time-dependent external forces, suffers a state of stress which is the superposition of two different aspects a time independent, long-term, behavior (sometimes improperly called hyperelastic ) opposed to a time dependent, short-term, response. Step-strain relaxation tests suggest that short term stresses are larger than the long term or quasi-static ones [117]. Moreover, oscillatoiy (sinusoidal) tests indicate that dissipative anelastic effects are significant, which leads to the consideration of a constitutive relation which depends not only on the current value of the strain but on the entire strain history. This assumption must be in accordance with some principles which restrict the class of rehable constitutive equations. These restrictions can be classified as physical and constitutive . The former are restrictirMis to which every rational physical theory must be subjected to, e.g., frame indifference. The latter, on the other hand, depends upon the material under consideration, e.g., internal symmetries. 

Let s analyze the relation between a so-called phase space correlation function and the conjugate response (susceptibility), which is a measure of the system s reaction to an externally imposed perturbation. Suppose one takes a system at equilibrium with Hamiltonian Hq and applies a time dependent external force starting at t=0, so that (McQuarrie,... 

A.3. Consider the 1-D motion of a point mass, connected to the origin by a harmonic spring, that experiences a frictional drag force and a time-dependent external force. The eqnation of motion is... 

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