The Representative Elementary Volume (REV) is a concept used primarily in the fields of materials science, geophysics, and hydrology. It refers to the smallest volume over which measurements can be taken so that the average properties of the material or medium are representative of the whole sample. The concept is crucial for understanding the macroscopic behavior of heterogeneous materials, such as soils, rocks, and composite materials.

Classical control theory is a framework for analyzing and designing control systems that operate in continuous time. It primarily deals with linear time-invariant (LTI) systems, where the behavior of the system can be described using ordinary differential equations. The main components of classical control theory include: 1. **System Modeling**: Classical control relies on mathematical models to represent dynamic systems. These models can be expressed in terms of transfer functions, which relate the input to the output of a system in the frequency domain.

Optimal control refers to a mathematical and engineering discipline that deals with finding a control policy for a dynamic system to optimize a certain performance criterion. The goal is to determine the control inputs that will minimize (or maximize) a particular objective, which often involves the system's state over time. ### Key Concepts of Optimal Control: 1. **Dynamic Systems**: These are systems that evolve over time according to specific rules, often governed by differential or difference equations.

In control theory, a compensator is a device or algorithm that modifies the behavior of a control system to improve its performance or stability. The purpose of a compensator is to enhance the system’s response to input changes, improve stability margins, reduce steady-state error, or shape the frequency response of the system.

The internal environment refers to the elements, factors, and conditions within an organization that can influence its operations, performance, and strategic direction. These elements are typically controllable and directly managed by the organization. Key components of the internal environment include: 1. **Organizational Structure**: This involves how the organization is arranged, including its hierarchy, roles, and communication channels.

Most applications of the Maxwell-Boltzmann distribution confirm the theory, but don't give a very direct proof of its curve.

Here we will try to gather some that do.

Measured particle speeds with a rotation barrel! OMG, pre electromagnetism equipment?

Is it the same as Zartman Ko experiment? TODO find the relevant papers.

Start by looking at: Maxwell-Boltzmann vs Bose-Einstein vs Fermi-Dirac statistics.

Bibliography:

This is not a truly "fundamental" constant of nature like say the speed of light or the Planck constant.

Rather, it is just a definition of our Kelvin temperature scale, linking average microscopic energy to our macroscopic temperature scale.

The way to think about that link is, at 1 Kelvin, each particle has average energy:per degree of freedom.

This is why the units of the Boltzmann constant are Joules per Kelvin.

For an ideal monatomic gas, say helium, there are 3 degrees of freedom. so each helium atom has average energy:

If we have 2 atoms at 1 K, they will have average energy $6/2k_{B}J$, and so on.

Another conclusion is that this defines temperature as being proportional to the total energy. E.g. if we had 1 helium atom at 2 K then we would have about $6/2k_{B}J$ energy, 3 K $9/2k_{B}J$ and so on.

This energy is of course just an average: some particles have more, and others less, following the Maxwell-Boltzmann distribution.

chemistry.stackexchange.com/questions/7696/how-do-i-distinguish-between-internal-energy-and-enthalpy/7700#7700 has a good insight:

Adds up chemical energy and kinetic energy.

Wikipedia mentions however that the kinetic energy is often negligible, even for gases.

The sum is of interest when thinking about reactions because chemical reactions can change the number of molecules involved, and therefore the pressure.

To predict if a reaction is spontaneous or not, negative enthalpy is not enough, we must also consider entropy via Gibbs free energy.

Bibliography:

I think these are the ones where $\Delta H\times \Delta S>0$, i.e. enthalpy and entropy push the reaction in different directions. And so we can use temperature to move the Chemical equilibrium back and forward.