# How do many conservative forces come together to make a non-conservative one?

Consider the mysterious force of friction. It drives our cars and bikes, slows them down, allows me to walk to the nearest dominoes store, allows me to eat my delicious pizza while ensuring that I stay sitting on the amazing seats provided by the pizza store.

Now, friction is a non-conservative force. We all know that the amount of work done in dragging a body across a carpet depends on the exact route you take. However, along with this practical experience, I have also been taught that most interactions between everyday objects are a consequence of electromagnetic force which is conservative in nature. So, my question is- how do several different Electromagnetic forces come together to make a new frictional force which is non-conservative in nature?

• It is all about how we can or cannot take all interactions into account. See for instance What exactly makes a force conservative? and its answers. Commented Jun 1, 2018 at 14:01

So my question how do several different Electromagnetic forces come together to make a new frictional force which is non-conservative in nature

Conservative forces (such as Coulomb forces) in the microscopic description do not form a non-conservative force on the same level of description. There is no friction force in the common sense on that level.

Electromagnetic forces complicate the question a bit since they are not conservative in the original sense - they can manifest delays or advanced effects and thus energy can get lost to or gained from outer space by radiation. This nonconservative property of EM forces is not commonly believed to be crucial to explain friction (irreversibility); instead it is believed widely that classically conservative forces such as impact forces acting on hard spheres and walls or gravitational/Coulombic forces already typically lead to irreversible evolution and thus to appearance of friction forces, provided the initial conditions are chosen "correctly".

So let me begin with the assumption that there are only conservative forces, such as the Coulomb forces, between the particles.

The friction force is usually thought of as macroscopic description of a phenomenon that is neglecting details of what happens on the microscopic scale.

If there are many mutually interacting particles (degrees of freedom), the calculations and simulations of the usual models show that there is, typically, macroscopically manifest transfer of energy from ordered, macro-visible forms (coordinated motion of particles, or concentration of particles in small space) into disordered, macro-invisible forms (chaotic thermal motion, dispersion of particle to the whole available space).

The easiest way to see this is simulation of motion of point particle gas, or more physical variant, a hard sphere gas in a container. It turns out that even while the interactions are conservative, the gas devolves towards a simple macroscopic state where density and average kinetic energy is the same throughout the container. Any initial macroscopic motion of the gas is damped down by apparent internal friction.

This typical behaviour is usually thought to be due to the large number of particles, but to be strictly correct, it does not happen for all possible initial conditions. If special initial conditions are chosen, one can observe "antifriction" - behaviour opposite to the idea of friction force, where the gas by itself gets from initial state into even more ordered state, for example, uniform gas can get spontaneously compressed and get moving as a whole in one direction. But this is believed to be very unlikely to happen if the initial conditions are "physical": this means they are chosen simultaneously to be:

1) consistent with macrostate we can prepare in a laboratory

2) otherwise microscopically random

Anti-friction has then so low probability that it is dismissed as impossible. That is why it is common to say that "microscopic conservative forces give rise to macroscopic friction forces", even though it is not completely accurate.

The fact that physical initial conditions almost always lead to physical "friction" behaviour, while there are equal number of conditions which would lead to "antifriction" behaviour is still topic of countless disputes and publications.

One interesting question this poses is: why do we live in the world where the macrostates we can prepare in the laboratory lead to "friction" behaviour, and not the other "antifriction" behaviour?

The traditional and accepted way to answer would be : that is how the world was set up in the beginning and from then on the conservative forces just propagated this deviation from symmetry to our time.

Another, less discussed view would be : the initial condition may not matter, the asymmetry may be because there are actually no conservative forces; the elementary forces are already slightly non-conservative - like retarded electromagnetic forces are. And this slight deviation from conservation is what would cause the physical behaviour to be always manifesting friction, not anti-friction.

The universe as a whole is conservative. But a particular subset of the universe (i.e. the system of interest) may not be if it is coupled to outside influence. Thus, forces are only non-conservative because they convert the energy and momentum from within the system of interest to outside of it. Friction represents the conversion of kinetic energy into heat and the transfer of momentum to objects outside the system of interest (e.g. when you model a car acceleration, you probably don’t care about the momentum of the pavement). So friction is non-conservative, but it wouldn’t necessarily be if we increased the scope of the modeled system.

• Friction does not necessarily mean energy is lost to the outside of the system; work or kinetic energy may be converted into internal energy while still remaining in the system. Commented Jun 1, 2018 at 20:55
• @JánLalinský Of course! It all comes down to how you define what is in the system and what is out. Commented Jun 1, 2018 at 23:17