# Phase transition in one dimension

• Avijeet
In summary, Susskind's lecture on statistical mechanics discusses the Ising model of magnetized spin systems and how there cannot be any phase transitions in one dimension due to the system's instability. At zero external field, the system has two macroscopic states - the ordered state with aligned spins and non-zero magnetization, and the disordered state with unaligned spins and zero magnetization. However, in one dimension, the ordered state is unstable to thermal fluctuations and will become demagnetized due to maximizing entropy. In higher dimensions, a competition between coupling energy and entropy allows for the ordered phase to exist at non-zero temperatures.
Avijeet
Hi,
I was listening to Susskind's lecture on statistical mechanics (lecture 8). He mentioned in relation to Ising model of magnetized spin systems that there could not be any phase transitions in one dimensions. He mentioned that it has to do with the stability of the system. Can anybody elaborate on this.

In the an Ising model (any dimension), there are two sorts of macroscopic states at zero external field: the ordered state, in which a finite fraction of the spins are aligned and the magnetization is non-zero, and a disordered state, in which the spins aren't aligned and the magnetization is zero. It turns out that in one dimension at zero external field, the ordered state is unstable to any thermal fluctuations. That means that flipping any spins at all due to a thermal fluctuation will cause all of the spins to fall out of alignment. Essentially, the free energy of the system at finite temperature is always minimized by maximizing the entropy - the coupling energy cannot overcome the entropy to cause the system to remain ordered. In two dimensions or higher this does not occur - you need to flip a finite fraction of the spins to cause the system to become demagnetized, as there is a competition between the coupling energy and entropy trying to minimize the free energy.

The ordered phase in one dimension (for zero external field) only exists at zero temperature. For non-zero field you can of course get some fraction of the spins to align with the field.

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## What is a phase transition in one dimension?

A phase transition in one dimension is a change in the physical state of a material or system as a result of a change in temperature, pressure, or other external factors. In one dimension, this means that the transition occurs along a single spatial dimension, rather than in three dimensions as in most traditional phase transitions.

## What causes a phase transition in one dimension?

Phase transitions in one dimension can be caused by a variety of factors, including changes in temperature, pressure, and external fields. These changes can affect the interactions between particles in the system, leading to a change in the system's physical state.

## What are the different types of phase transitions in one dimension?

There are several types of phase transitions that can occur in one dimension, including continuous/second-order transitions, discontinuous/first-order transitions, and topological transitions. Continuous transitions involve a gradual change in the system's physical properties, while discontinuous transitions involve a sudden change. Topological transitions occur when the topology of the system changes, such as when a loop or knot appears in the system.

## What are the applications of studying phase transitions in one dimension?

Studying phase transitions in one dimension has many practical applications, including understanding the behavior of materials at different temperatures and pressures, developing new materials with unique properties, and designing efficient energy storage systems. It also has implications in fields such as condensed matter physics, materials science, and engineering.

## How do scientists study phase transitions in one dimension?

Scientists use various experimental and theoretical methods to study phase transitions in one dimension. These include techniques such as X-ray diffraction, neutron scattering, and computer simulations. By analyzing the behavior of the system at different temperatures and pressures, scientists can determine the nature and properties of the phase transition and better understand the underlying physics.

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