There is an one-dimensional lattice with lattice constant a...

There is an one-dimensional lattice with lattice constant $a$ as shown in Fig. 2.1. An atom transits from a site to a nearest-neighbor site every $\tau$ seconds. The probabilities of transiting to the right and left are $p$ and $q=1-p$ respectively. FIGURE CAN'T COPY. (a) Calculate the average position $\bar{x}$ of the atom at the time $t=N \tau$, where $N \gg 1$; (b) Calculate the mean-square value $\overline{(x-\bar{x})^2}$ at the time $t$.

There is an one-dimensional lattice with lattice constant $a$ as shown in Fig. 2.1. An atom transits from a site to a nearest-neighbor site every $\tau$ seconds. The probabilities of transiting to the right and left are $p$ and $q=1-p$ respectively.
FIGURE CAN'T COPY.
(a) Calculate the average position $\bar{x}$ of the atom at the time $t=N \tau$, where $N \gg 1$;
(b) Calculate the mean-square value $\overline{(x-\bar{x})^2}$ at the time $t$.

Key Concepts

One-Dimensional Random Walk

This concept involves a particle moving along a single line where its position changes are determined by random steps. Each step is independent of the previous ones, and the process is typically modeled as a sequence of moves that can be either to the left or right, often with specified probabilities. This forms the basis of many stochastic processes and has applications in statistical physics, chemistry, and finance.

Transition Probabilities

Transition probabilities refer to the likelihood of the particle moving from one site to the next. In the context of random walks, these probabilities determine how often the particle moves in a particular direction (for example, left or right). They are central to defining the behavior of the walk, as they can introduce a bias or drift when the probabilities for different directions are not equal.

Drift in Random Walks

Drift describes the net directional movement that can occur when the probabilities for moving in different directions are unbalanced. In a random walk with bias, such as when one transition probability exceeds the other, the particle will tend to move more in one direction on average over many steps. This concept is important in understanding how systematic trends emerge from random processes.

Mean Displacement

Mean displacement is the average position of the particle after a given number of steps. It is calculated by taking into account the individual contributions of each step, weighted by the direction and the step length. In biased random walks, the mean displacement will reflect the net effect of the unequal transition probabilities, indicating a shift away from the initial position over time.

Mean-Square Displacement

Mean-square displacement is a measure of the spread or dispersion of the particle positions around the mean displacement. It quantifies how far, on average, the particle is from its expected position after a given number of steps. This concept is crucial for understanding diffusion processes and provides insight into the variability and uncertainty inherent in stochastic systems.

Lattice and Lattice Constant

In the context of random walks, a lattice refers to the discrete set of positions where the particle can reside, and the lattice constant is the fixed distance between these positions. This spatial structure provides the framework for analyzing movements in systems with regular, quantized steps, and it is particularly relevant in solid-state physics and crystallography.

Transcript

Please give Ace some feedback