Single-file systems particles

Mean time between two jump attempts, residence time Mean exchange time between two adjacent molecules Average time that a particle which is foimd at site i, has already spent in the single-file system... 

Residence time distribution fimction (of a particle at site i in the single-file system)... 

Since normal and single-file diffusion are described by the same propagator, Eq. 5, due to the analogy of Eq. 2 and Eq. 3 the propagation pattern of a given particle in a single-file system coincides with that of normal diffusion... 

The great expense in calculation time due to the inevitably large particle numbers in single-file systems calls for the application of simplified potentials. Figure 1 shows the results obtained for spherical molecules diffusing in an unstructured tube [22]. Particle-particle and particle-wall interactions have been simulated by a shifted-force Lennard-Jones potential [26] and an... 

The concept of single-file diffusion has most successfully been applied for MD simulations in carbon nanotubes [36-39], yielding both the square-root time dependence of the molecular mean square displacement and a remarkably high mobility of the individual, isolated diffusants. In [40-42], the astonishingly high single-particle mobilities in single-file systems have been attributed by MD simulations to a concerted motion of clusters of the adsorbed molecules. 

As soon as the movements of different species have to be distinguished from each other, however, the mutual correlation of the molecules in singlefile systems makes it impossible to predict the evolution of the particle distributions by differential equations. Eor this reason, the time dependence of the tracer exchange in single-file systems has thus far only been investigated by Monte Carlo simulations [1,55-57]. 

Fig.4 Comparison of the concentration profiles of tagged particles obtained by DMC simulations for tracer exchange in single-file systems of length L (oscillating solid lines) with the concentration profiles for normal diffusion, with Dsim and N given in Table 1 (solid lines) at times ti = 0.93 x 10 r, t2 = 2.1 x 10 r, t = 3.7 x 10 , and t4 = 7.6 x 10 r (r is the duration of the elementary diffusion step). From [57] with permission...

Our understanding of diffusion and reaction in single-file systems is impaired by the lack of a comprehensive analytical theory. The traditional way of analytically treating the evolution of particle distributions by differential equations is prevented by the correlation of the movement of distant particles. One may respond to this restriction by considering joint probabilities covering the occupancy and further suitable quantities with respect to each individual site. These joint probabilities may be shown to be subject to master equations. 

Fig. 8 Average residence time profile (in units of the time r between two jump attempts) of the particles in the single-file systems considered in Fig. 7. From [72] with permission...

The benefit of the analytical treatment presented thus far for the calculation of the characteristic functions of the single-file system is only limited by the increasing complexity of the joint probabilities and the related master equations. This treatment, however, has suggested a most informative access to the treatment of systems subjected to particle exchange with the surroundings and to internal transport and reaction mechanisms [74,75]. Summing over all values (Ji = 0 and 1 and, subsequently, over all sites i, Eq. 31 may be transferred to the relation Eq. 34... 

The difference between the correlation functions in a canonical and a grand canonical ensemble is largest for one-dimensional systems with hard core interactions. This occurs because in such a system the particles cannot pass each other. This also leads to a significantly different diffusion behavior (single file diffusion). In a grand canonical setting, however, particles can pass each other via the... 

Exact solutions. It is possible to obtain some exact results for mean residence times even for channels with large numbers of particles although the results are typically cumbersome [90, 91]. Here, we briefly sketch the main points of the derivation for the case of single-file transport in a uniform channel in equilibrium with a solution of particles [90]. Most generally, the system of multiple particles in a channel is described by the multi-particle probability function P(x,t y) that the vector of particles positions is x at time t, starting from the initial vector y [53, 90, 92]. The crucial insight is that because the particles cannot bypass each other, the initial order of the particles is conserved if y < y for any two particles at the initial time, it implies that x < for all future times. That is, the parts of the phase space accessible to these particles are bounded by the planes defined by the condition = x in the vector space x. This implies a reflective boundary condition at the x = plane for any two different particles m and n,... 

Since we are interested in cellular activities where several motors could be involved, a simplified mean-filed model emerge with the assumption that motors a) work independent of each other b) share the applied load equally. These equations are able to predict the transport properties of cargo particles such as their effective velocity and average run length. Two most important quantities that need to be mentioned here are the force-velocity relation and detachment rate of motor when a load is applied to the cargo. Several experiments have shown that velocity Vn(F) of a single motor in system of n bound motors decreases almost linearly with the force F applied against the motor movement. 

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