Gaussian random potential

The operator V l corresponding to V being now well defined, we shall associate a Gaussian random potential x(r) with the chains. To introduce this potential, we start from the trivial identity... 

A Gaussian random potential with short range correlations. 

We now revisit the meaning of the term localization as applied to polymers in a random medium. Although some authors connect the compact size of the chain when L oo with the notion of localization, this is actually not so. The compact size should be viewed as a separate feature from the notion of localization. Recall that for a Gaussian chain in an uncorrelated Gaussian random potential of variance g the chain has typical size... 

The transition occurs between v = 0.041 and 0.042. Again we could find analytically almost the entire solution both for w x and at the transition. The solution is given in the Appendix. There we show that the transition occurs at w = 0.04142. We observe that as for the case of a Gaussian random potential the ratio w/m changes from 1 to Vi. We also checked that the free energy from Table II is lower than what one would obtain by constraining F to be in Region I, i.e. F < Vi. 

We then proceeded to discuss the more realistic case of a chain embedded in a sea of hard obstacles. Here, we showed that the chain size exhibits a rich scaling behavior, which depends critically on the volume of the system. In particular, we showed that a medium of hard obstacles can be approximated as a Gaussian random potential only for small system sizes. For larger sizes a completely different scaling behavior emerges. 

Thermal disorder merely adds an additional Gaussian random potential whose variance adds linearly to w, as already discussed. 

The well-known result of Halperin and Lax for the tail of the density of an electron moving in a Gaussian random potential is reviewed. 

Here, 7 is the friction coefficient and Si is a Gaussian random force uncorrelated in time satisfying the fluctuation dissipation theorem, (Si(0)S (t)) = 2mrykBT6(t) [21], where 6(t) is the Dirac delta function. The random force is thought to stem from fast and uncorrelated collisions of the particle with solvent atoms. The above equation of motion, often used to describe the dynamics of particles immersed in a solvent, can be solved numerically in small time steps, a procedure called Brownian dynamics [22], Each Brownian dynamics step consists of a deterministic part depending on the force derived from the potential energy and a random displacement SqR caused by the integrated effect of the random force... 

The force arising from the potential is F, while R is a gaussian random force. The net effect of the collisions , i.e. dynamical interactions between the particle and solvent molecules, is thus approximately accounted for by the frictional, or damping force, Fj. = —fiC,x, where is a friction constant related to the time correlation of the random force ... 

Here, m is the mass of the particles, V(r) the potential energy of the system, 7 the friction constant, and. F is a Gaussian random force uncorrelated in time that satisfies the fluctuation dissipation theorem [20]... 

To generate the trajectories that result from stochastic equations of motion (14.39) and (14.40) one needs to be able to properly address the stochastic input. For Eqs (14.39) and (14.40) we have to move the particle Linder the influence of the potential T(.v), the friction force—yvm and a time-dependent random force R(t). The latter is obtained by generating a Gaussian random variable at each time step. Algorithms for generating realizations of such variables are available in the applied mathematics or numerical methods hterature. The needed input for these algorithms are the two moments, (2J) and In our case (7 ) = 0, and (cf. Eq. (8.19)) = liiiyk/jT/At. where Ai is the time interval... 

A quenched random medium, such as a rough surface or a frozen gel network, is a complex structure that can in principle be modelled by a complicated potential function y(R). However, we will not be interested in the physical properties of a polymer chain immersed in a specific environment, but rather in an ensemble of similar environments. Hence, we will have to specify instead the probability distribution of the random potential y(R). Here, we will consider random potentials that are taken fi om a Gaussian distribution defined by... 

In order understand the conformational statistics of a Gaussian chain in a random potential, we map the partition sum to an imaginary time Schrodinger eqnation. This mapping (see Ref. [7] Eqs. (3.12)-(3.18)) is given by... 

The above results show that the 1-step RSB solution correctly predicts some important features of the eigenvalue distribution. More importantly, we have shown that the 1-step RSB solution can be interpreted in terms of the eigenstates of the Schrodinger equation with a random potential. However, there are differences and these reveal the limitations of the 1-step RSB solution. For example, all the localized states are approximated by the same Gaussian profile when in fact the localization lengths should increase with energy. 

The result for the radius of gyration of the chain, as represented by Rmi is the same result as for the case of the Gaussian distributed random potential, but with the strength g replaced by x — x). The polymer in this case is localized and its size is independent of L for large L. 

Consider first the case of a random potential with a Gaussian distribution. For simplicity, the discussion in the rest of this section be limited to three spatial dimensions (d = 3). Recall that in the case when there is no self-avoiding interactions the optimal size of a chain is found by minimizing the free energy F in Eq. 23. This yields... 

In this chapter we have demonstrated the rich behavior of polymer chains embedded in a quenched random environment. As a starting point, we considered the problem of a Gaussian chain free to move in a random potential with short-ranged correlations. We derived the equilibrium conformation of the chain using a replica variational ansatz, and highlighted the crucial role of the system s volume. A mapping was established to that of a quantum particle in a random potential, and the phenomenon of localization was explained in terms of the dominance of localized tail states of the Schrodinger equation. 

Our first application of the Gaussian Approximation Potentials was a set of potentials for simple semiconductors. We calculated the total energies and forces of a number of configurations, which were generated by randomly displacing atoms in the perfect diamond structure. We included 8-atom and 64-atom supercells at different lattice constants and we perturbed the lattice vectors in some cases. The atoms were displaced at most by 0.3 A. 

The gradual increase of Rq ) in Fig. 8.22 corresponds to a gradual evolution of the molecular shape from nearly two-dimensional structures near the walls to three-dimensional Gaussian random-coils at a distance from the walls close to the unperturbed radius of gyration. The molecular shape of chains with center of mass close to the walls, when examined in details, can be described as an apparently random combination of train and loop sequences. The average length of train sequences (that is, a sequence of beads located in the first layer of thickness a near a wall) is 4.1 beads for chains of 100 beads in the absence of a bending potential, and increases to 4.6 beads for the polyethylene-like bead chains simulated in system E, which corresponds to approximately 16 methylene units. Hence, one should take... 

In the presence of a potential U(r) the system will feel a force F(rj,) = — ViT/(r) rj,. There will also be a stochastic or random force acting on the system. The magnitude of that stochastic force is related to the temperature, the mass of the system, and the diffusion constant D. For a short time, it is possible to write the probability that the system has moved to a new position rj,+i as being proportional to the Gaussian probability [43]... 

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