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Linear Viscoelasticity
Published in Timothy P. Lodge, Paul C. Hiemenz, Polymer Chemistry, 2020
Timothy P. Lodge, Paul C. Hiemenz
We have inserted a front factor, Gp, which gives the amplitude of each mode; Gp must have the units of a modulus. We now invoke the equipartition theorem of statistical mechanics: each degree of freedom, or normal mode, acquires kT of thermal energy. Furthermore, we can assume that the modulus will increase linearly with the number of chains per unit volume because each chain can store the same amount of elastic energy under deformation. The number of chains per unit volume is cNav/M, where c is the concentration in g/mL. Therefore we can equate Gp with (cNav/M) kT = cRT/M:G(t)=cRTM∑p=1Nexp(−t/τp)
Statistical Physics and Thermodynamics Primer
Published in Thomas M. Nordlund, Peter M. Hoffmann, Quantitative Understanding of Biosystems, 2019
Thomas M. Nordlund, Peter M. Hoffmann
What is a “distinct mode” (or “degree of freedom” in some books)? For our purposes a mode can contain ½kBT of energy if it is a dynamic variable on which contained energy is quadratically dependent. Examples include ½mv2 and ½kx2. The Equipartition theorem applies to any system of many particles, except sometimes at low temperature, where the time to reach equilibrium may be longer than the observation time.
Interlude of Physics II: Thermodynamics
Published in Franco Battaglia, Thomas F. George, Understanding Molecules, 2018
Franco Battaglia, Thomas F. George
What has been found for an ideal gas of structureless particles, and applied to electrons, may be applied to atomic gases as well. The equipartition theorem validity condition (9.61) still holds, but the typical concentrations of atomic gases are of the order of 103 smaller and the typical masses are of the order of 104 greater than those of conduction electrons; hence, at the resulting characteristic temperature below which the gas should be treated quantum mechanically, the atomic species is surely in its condensate phase. In other words, at all temperatures at which an atomic species is in the gas phase, the gas may be treated classically and the equipartition theorem is applicable.
Nonequilibrium Green’s functions (NEGF) in vibrational energy transport: a topical review
Published in Nanoscale and Microscale Thermophysical Engineering, 2021
Another atomistic simulation tool to describe vibrational energy transport is Molecular dynamics (MD) [53–55]. This method is well-suited for energy transport in systems at high-temperatures (above room temperature) or in systems where a truncated series expansion of their potential energy is not a good approximation. MD is the default method of choice to study vibrational energy transport in disorder materials. MD is a purely classical approach that finds the time evolution of the atomic positions by solving Newton’s equations of motion for each atom. This procedure distributes the energy equally among degrees of freedom in the system (Equipartition theorem), which deviates from reality as the temperature decreases. Most of the existing MD implementations to study vibrational energy transport rely on empirical interatomic potentials to describe atomic interactions, which restrict the predictability of the method. Coupling MD with first-principles calculations is a powerful tool being used in many fields of chemistry and materials science [56, 57]. However, computational constraints on the size of the simulation domain have limited its applicability to vibrational energy transport studies.
Application of agent-based paradigm to model corrosion of steel in concrete environment
Published in Corrosion Engineering, Science and Technology, 2018
The movement (forcing them to randomly move in the solution) of the ions, i.e. chlorides and hydroxides, in the model were used to simulate the mobile nature of these ions in the solution. The speed of this movement characterised by the ambient temperature and it was calibrated, using the equipartition theorem, to represent the laboratory temperature (i.e. 23°C). The equipartition theorem is a useful guide to the average energy associated with each degree of freedom when the sample is at a temperature, T. According to the equipartition theorem, the mean energy of each atom or molecule in thermal equilibrium can be calculated using the following equation [40,41]:where v and m are velocity and mass, respectively. Thus, by knowing the mass of each ion in the solution, the velocity of the ions in the solution (agents’ movements in the model) can be adjusted to correspond to the predefined temperature, i.e. 23°C.