Statistical Mechanics Lecture 9

The speaker begins by addressing a previous misunderstanding about the Ising model, clarifying that Ising did solve the one-dimensional model correctly but erred in assuming no phase transition exists in any dimension. The lecture aims to explore phase transitions in the Ising model, particularly in higher dimensions, using an approximation method. The concept of phase transition as a sudden change in system properties, notably with temperature variation, is introduced. The simple example of a collection of magnets with an energy function based on their 'up' or 'down' state is revisited, and the notation is simplified using the variable Sigma and the product J.

The paragraph focuses on calculating the partition function for a single spin, which is simplified by considering the system's independence. The partition function is expressed as a hyperbolic cosine function of beta times J. The energy of a single spin is derived, and the concept of averaging the energy is discussed. The probability of a spin being up or down is also explored, leading to the introduction of the hyperbolic tangent function as a descriptor of the spin's average state.

The discussion shifts to the one-dimensional Ising model, where the energy depends on neighboring spins. The total energy is the sum of products of neighboring spins, each multiplied by a constant J. The ground state of the system is explored, revealing two possible configurations with all spins aligned 'up' or 'down'. The impact of changing the sign of J is also considered, showing that the system remains mathematically identical despite appearing different.

The concept of correlation functions is introduced to measure the influence one spin has on another further down the chain. The speaker uses the analogy of a game of telephone to illustrate how the signal or influence of a spin being up can propagate through the system. The importance of the partition function in calculating these probabilities is emphasized, and the challenge of summing over all possible configurations is acknowledged.

The speaker discusses the diagnostic value of the average product of spins at different locations to understand the influence of one spin on another and its propagation through the system. The game of telephone analogy is revisited to explain how memory or influence fades with distance. The concept of 'memory' in the context of a spin's influence on its neighbors is explored, and the mathematical implications of this fading influence are detailed.

A strategic approach to understanding the Ising model is presented, which involves focusing on the links between spins rather than the spins themselves. This approach simplifies the partition function and allows for a clearer understanding of the energy associated with the bonds between neighboring spins. The speaker also introduces the concept of bond variables and how they can be used to express the partition function.

The mean field approximation is introduced as a method to analyze the Ising model in high dimensions. The approximation assumes that each spin is influenced by a field created by its many neighbors, leading to a self-consistent field approximation. The speaker derives an equation for the average spin and discusses how this approximation can predict a phase transition at high temperatures, where the system exhibits magnetization.

The speaker explains the phenomenon of magnetization in the context of the Ising model, highlighting it as a phase transition where the system develops a net magnetization. The role of temperature in this transition is discussed, with the system exhibiting random behavior at high temperatures and a preference for alignment at lower temperatures. The speaker also touches on the implications of adding a small magnetic field to the system.

The discussion shifts to the behavior of the Ising model in higher dimensions, with the speaker explaining that the model behaves differently in two dimensions compared to one. The concept of dimensional dependence is explored, and the speaker suggests that the model's behavior in high dimensions can be understood through approximations and limits, even if it cannot be solved directly.

The energy costs associated with flipping spins in the Ising model are calculated, with comparisons made between one-dimensional and two-dimensional cases. The speaker illustrates how the number of broken bonds and the resulting increase in energy depend on the number of flipped spins and the dimensionality of the system. The stability of the system with respect to spin flipping is also discussed.

The speaker concludes the discussion by acknowledging the complexity of the Ising model and the richness of its implications for understanding magnetism. They express an intention to continue exploring topics related to magnets and possibly metals in future lectures. The speaker also invites the audience to visit Stanford.edu for more information.