# In the context of statistical mechanics can anyone define temperature?

• Monsterboy
In summary, temperature is not solely based on the average kinetic energy of particles in a system. In statistical physics, temperature is defined as the inverse of the derivative of entropy with respect to total energy, where entropy is proportional to the logarithm of the number of states with that energy. This definition takes into account the total energy of a system, including potential energy from interactions. This allows for the definition of temperature in systems that do not involve motion. In systems where the total energy is the sum of kinetic and potential energy, there is a relationship between the two, known as the virial theorem. Therefore, when discussing average kinetic energy, we are also indirectly discussing the total energy. However, it is the total energy that gives rise to the

#### Monsterboy

I was told that defining temperature as the "average kinetic energy of the particles in a system" is not accurate enough.

Well, we have come to have a more general sense of temperature in statistical physics. It has to do with total energy instead of kinetic energy. We define temperature as

$$\frac{1}{T} = \left( \frac{\partial S(E)}{\partial E}\right)_{V, N}$$

Where ##S(E)## is the entropy the system when it has total energy E and the ##\left. \right)_{V, N}## means hold the volume and number of particles constant while taking the derivative.
The entropy (of an ergotic system in equilibrium) is in turn proportional to the log of the number of states with that energy, ##\Omega(E)##
$$S(E) = k_B \ln \Omega(E)$$
This is also can be expressed in terms of a more general formula
$$S(E) = -k_B \sum_i p_i \ln p_i$$
Where the sum is over micro-states and ##p_i## is the probability for microstate i.

So, one important difference from "average kinetic energy" is that this definition involves the total energy, not just kinetic. Potential energy from interactions gets a role in determining temperature.

So for example we can consider the "atoms" in a magnet. Depending on the temperature the "atoms" that make up the magnet have a differing tendency to align with an external magnetic field. An important part of this temperature has nothing to do with motion. Instead it has to do with the different possible alignment configurations. So, the statistical physics framework allows us to define a temperature for practically any system that has a total energy, even those that don't involve motion.

In systems where the total energy is the sum of kinetic plus a potential energy that is a function of the positions, there tends to be a relationship between the kinetic energy and the total energy. This is called the virial theorem. For example average kinetic energy can be proportional to the total energy. So for these types of systems, when we say something about the average kinetic energy, we are also saying something about the total energy as well.

But it's really from the total energy that the statistical physics sense of temperature arises.

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Monsterboy

## 1. What is temperature?

Temperature is a measure of the average kinetic energy of the particles in a substance. It is a fundamental concept in statistical mechanics and is closely related to the concept of thermal equilibrium.

## 2. How is temperature related to statistical mechanics?

In statistical mechanics, temperature is defined as the derivative of the energy with respect to entropy. This relationship helps us understand how the microscopic properties of a system (such as the motion of particles) influence its macroscopic behavior (such as temperature).

## 3. How is temperature measured?

Temperature can be measured using various instruments such as thermometers, which use the expansion and contraction of a liquid or gas to indicate temperature. In statistical mechanics, temperature is measured in units of energy, such as joules or electron volts.

## 4. How does temperature affect the behavior of particles in a substance?

As temperature increases, the average kinetic energy of particles also increases. This can lead to changes in the state of matter, such as melting or boiling, and can affect the rate at which chemical reactions occur. In statistical mechanics, temperature is also related to the probability of a particle having a certain energy level.

## 5. What is absolute temperature?

Absolute temperature is a temperature scale that is based on the properties of an ideal gas. It is measured in units of kelvin (K) and is defined as 0 K at absolute zero, the point at which all molecular motion ceases. Absolute temperature is often used in statistical mechanics as it allows for a more precise understanding of the behavior of particles in a substance.