System of N classical non-interacting particles

In a system of N classical non-interacting particles, each particle is considered to be independent of the others and does not interact with them. This means that the particles do not influence each other's motion or properties, such as energy and temperature.

External potential in 3 dimensions

The external potential is the potential energy associated with the interaction of the particles with their surroundings. In this case, the external potential is given in three dimensions, implying that the particles are confined in a three-dimensional space.

Internal energy of the system

The internal energy of a system is the total energy contained within it. In the case of a system of classical non-interacting particles, the internal energy is the sum of the kinetic energies and potential energies of all the particles in the system.

The kinetic energy of each particle can be described by the classical formula:

K.E. = (1/2)mv^2

Where m is the mass of the particle and v is its velocity.

The potential energy of each particle depends on the external potential. Since the particles are non-interacting, the potential energy for each particle can be considered separately. The potential energy is given by:

P.E. = U(x, y, z)

Where U is the potential energy function that depends on the position of the particle in the three-dimensional space.

The total internal energy of the system is then the sum of the kinetic energies and potential energies of all the particles:

Internal Energy = Σ(K.E. + P.E.)

Temperature T

The temperature T of the system is a measure of the average kinetic energy of the particles. In classical mechanics, the temperature is related to the average kinetic energy by the equation:

K.E. = (3/2)kT

Where k is the Boltzmann constant.

Final Words

In summary, in a system of N classical non-interacting particles with a given external potential in three dimensions, the internal energy of the system is the sum of the kinetic energies and potential energies of all the particles. The kinetic energy is related to the temperature of the system, while the potential energy depends on the particular form of the external potential.