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Austrian physicist Ludwig Boltzmann explained entropy as the measure of the number of possible microscopic arrangements or states of individual atoms and molecules of a system that comply with the macroscopic condition of the system. He thereby introduced the concept of statistical disorder and probability distributions into a new field of thermodynamics, called statistical mechanics, and found the link between the microscopic interactions, which fluctuate about an average configuration, to the macroscopically observable behaviour, in form of a simple logarithmic law, with a proportionality constant, the Boltzmann constant, that has become one of the defining universal constants

in any natural process there exists an inherent tendency towards the dissipation of useful energy. ( Lazare Carnot in an 1803 paper Fundamental Principles of Equilibrium and Movement)

Lazare’s son, Sadi Carnot, published Reflections on the Motive Power of Fire, which posited that in all heat-engines, whenever “caloric” (what is now known as heat) falls through a temperature difference, work or motive power can be produced from the actions of its fall from a hot to cold body. He used an analogy with how water falls in a water wheel.

the nature of the inherent loss of usable heat when work is done (1850s and 1860s, German physicist Rudolf Clausius) a dissipative use of energy. From the prefix en-, as in ‘energy’, and from the Greek word τροπή [tropē], which is translated in an established lexicon as turning or change[7] and that he rendered in German as Verwandlung, a word often translated into English as transformation, in 1865 Clausius coined the name of that property as entropy.[8] The word was adopted into the English language in 1868.

In 1877, Boltzmann visualized a probabilistic way to measure the entropy of an ensemble of ideal gas particles, in which he defined entropy as proportional to the natural logarithm of the number of microstates such a gas could occupy. The proportionality constant in this definition, called the Boltzmann constant. Henceforth, the essential problem in statistical thermodynamics has been to determine the distribution of a given amount of energy E over N identical systems

The concept of entropy is described by two principal approaches, the macroscopic perspective of classical thermodynamics, and the microscopic description central to statistical mechanics. The classical approach defines entropy in terms of macroscopically measurable physical properties, such as bulk mass, volume, pressure, and temperature. The statistical definition of entropy defines it in terms of the statistics of the motions of the microscopic constituents of a system — modelled at first classically, e.g. Newtonian particles constituting a gas, and later quantum-mechanically (photons, phonons, spins, etc.). The two approaches form a consistent, unified view of the same phenomenon as expressed in the second law of thermodynamics, which has found universal applicability to physical processes.

entropy has the dimension of energy divided by temperature, and the unit joule per kelvin (J/K). In any process, where the system gives up $\vartriangle{E}$ of energy to the surrounding at the temperature $T$, its entropy falls by $\vartriangle{S}$ and at least $T\cdot\vartriangle{S}$ of that energy must be given up to the system’s surroundings as a heat. Otherwise, this process cannot go forward. In classical thermodynamics, the entropy of a system is defined if and only if it is in a thermodynamic equilibrium

Dimensionless quantities, or quantities of dimension one,[1] are typically expressed as ratios that align with another system, these quantities do not necessitate explicitly defined units. The number one is recognized as a dimensionless base quantity.[4] Radians serve as dimensionless units for angular measurements, derived from the universal ratio of 2π times the radius of a circle being equal to its circumference.[5]

There have been periodic proposals to “patch” the SI system to reduce confusion regarding physical dimensions. For example, a 2017 op-ed in Nature[11] argued for formalizing the radian as a physical unit. The idea was rebutted[12] on the grounds that such a change would raise inconsistencies for both established dimensionless groups, and for mathematically distinct entities that happen to have the same units, like torque (a vector product) versus energy (a scalar product).

Torque is the rotational analogue of linear force.¹ It is also referred to as the moment of force (also abbreviated to moment). The symbol for torque is typically $\tau$, the lowercase Greek letter tau.

[!quote]  James Thomson appeared in print in April, 1884.
Just as the Newtonian definition of force is that which produces or tends to produce motion (along a line), so torque may be defined as that which produces or tends to produce torsion (around an axis).

A force applied perpendicularly to a lever multiplied by its distance from the lever’s fulcrum (the length of the lever arm) is its torque. Therefore, torque is defined as the product of the magnitude of the perpendicular component of the force and the distance of the line of action of a force from the point around which it is being determined. In three dimensions, the torque is a pseudovector; for point particles, it is given by the cross product of the displacement vector and the force vector. The direction of the torque can be determined by using the right hand grip rule: if the fingers of the right hand are curled from the direction of the lever arm to the direction of the force, then the thumb points in the direction of the torque.[8] It follows that the torque vector is perpendicular to both the position and force vectors and defines the plane in which the two vectors lie. The resulting torque vector direction is determined by the right-hand rule. Therefore any force directed parallel to the particle’s position vector does not produce a torque.[9][10] The magnitude of torque applied to a rigid body depends on three quantities: the force applied, the lever arm vector[11] connecting the point about which the torque is being measured to the point of force application, and the angle between the force and lever arm vectors. In symbols:

$\mathbf{\tau = r \times F} \Rightarrow \tau = rF_{\bot} = rF\sin{\theta}$

where

• $\tau$ is the torque vector and $\tau$ is the magnitude of the torque,
• $\mathbf{r}$ is the position vector (a vector from the point about which the torque is being measured to the point where the force is applied), and $r$ is the magnitude of the position vector,
• $\mathbf{F}$ is the force vector, $F$ is the magnitude of the force vector and $F_{\bot}$ is the amount of force directed perpendicularly to the position of the particle,
• $\times$ denotes the cross product, which produces a vector that is perpendicular both to $\mathbf{r}$ and to $\mathbf{F}$ following the right-hand rule,
• $\theta$ is the angle between the force vector and the lever arm vector.

The SI unit for torque is the newton-metre (N⋅m).