# Definition of Fluctuations and Perturbations

The terms fluctuations and perturbations are frequently used in physics with different meanings. But they are confusing. Both terms seems to be same. Is there any one who can explain lucidly these terms and show difference between these two terms?

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In my experience, perturbation is most often used for something where perturbation theory (perturbation series) is applicable. en.wikipedia.org/wiki/Perturbation_theory – mtrencseni Mar 2 '12 at 22:21

Fluctuation

Suppose you repeatedly measure pressure $P$ in a piston due to gas in equilibrium. If your measurement apparatus is of high quality, you will read off numbers that fluctuate around a mean value $P_0$. Pressure, which is force divided by area, $P=F/A$, is due to gas molecules hitting the piston, and as the number of collisions and velocities vary, you'll have tiny fluctuations in your measurement.

Perturbation

Back in school we usually used the word perturbation to refer to perturbation theory (PT). PT is a way to model the dynamics of a physical system in a first, linear approximation. What we'd usually do is imagine that the system is in a stable equilibrium, i.e. it's sitting at the bottom of an energy valley. Suppose that the relevant parameter describing the system is $x$, its value for the energy minimum is $x_0$, and $E(x_0)=E_0$. You "perturb" the system, which means you imagine that the system moved away - for whatever reason - a little bit from the energy minimum; let $\Delta x = x_0 - x$. You can construct a perturbation series of energy in powers of $\Delta x$, like this: $E = E_0 + E_1 \Delta x + E_2 \Delta x^2 + ...$. Since we are perturbing around an energy mimimum, $E_1$ would be zero and $E_2 > 0$. You disregard higher order terms ($\Delta x^3$ and higher) because $\Delta x$ is small, and you end up with something like a spring, because $E \approx \Delta x^2$ in this approximation. This is linear, because the force law in this approximation is linear, as $F = - \frac{dE}{dx} = - 2 E_1 \Delta x$. What you end up with, in this first approximation, is oscillation around $x_0$.

An example that comes to mind is the oscillation of small test bodies like a satellite around stable Lagrange points of the Sun-Earth system (quotes from the linked Wikipedia page):

"When a body at these points is perturbed, it moves away from the point, but the factor opposite of that which is increased or decreased by the perturbation (either gravity or angular momentum-induced speed) will also increase or decrease, bending the object's path into a stable, kidney-bean-shaped orbit around the point..."

and

"The L4, and L5 positions are stable as a ball at the bottom of a bowl would be stable: small perturbations will move it out of place, but it will drift back toward the center of the bowl."

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A perturbation is a small change (usually deterministic and known), while a fluctuation is a (not necessarily small) random perturbation with mean zero (and therefore either unknown or unrepeatable).

Usually one talks about a perturbation in the context of perturbation theory. Perturbation theory is used to study a system that is slightly different from a nice system where you can do calculations easily; the difference from the nice system is the perturbation.

On the other hand, one talks about fluctuations when deriving results for the mean of certain quantities; the fluctuations are the deviations from the mean. Another common word for fluctuations is noise (used in the singular only). In realizations of stochastic processes, fluctuations are visible as contributions of large but irregular frequency, whereas a perturbation would be in this context a small change of a parameter of the system or the forcing term.

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I would not pretend to be definitive on this, but I would take Fluctuations most often to refer to stochastic deformations of a $\mathit{particular}$ signal, whereas Perturbations most often refer to deformations (which can be either deterministic or stochastic) of a $\mathit{general}$ dynamics (which describes the time evolution of many different signals of a given type).

These words are appropriated in different ways by different authors, however, so that ultimately the sense intended has to be deduced from the mathematical constructions used.

I will be interested to see other people's take on the difference.

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OK answer. One could also discuss oscillations and corrections in the same thread. There are various differences. Oscillations are small numbers (a bit bigger than fluctuations) around the "normal level" which are time-dependent. The time dependence sort of applies to fluctuations, too: fluctuations should be time-dependent. On the other hand, perturbations are changes made to the laws of physics, "before" we "kickstart" time evolution. Corrections are like perturbations except that perturbations should be corrections to a solvable/understood zeroth approximation, not a generic one. And so on. – Luboš Motl Jan 25 '12 at 15:20
Lobus MotI Answer makes sense but I am still confused anout these terms perturbations and fluctuation – Abdur Rasheed Jan 30 '12 at 5:23