_critical exponents_ are parameters that are said to be _[universal](universality%20of%20critical%20phenomena)_ that describe how physical quantities behave close to the [[critical point]]. This universality of critical exponents is observed by noticing that there are a few sets of critical exponents that are found universally in unrelated physical systems. However, we find different numerical values for these exponents for different systems. ^f766fe # deriving critical exponents In order to derive a relation for critical exponents, consider a function $f(t)=f_s(t)+f_r(t)$ where $f(t)>0$ for $t>0.$ Here $f_r(t)$ is a [Regular functions](Regular%20functions.md) and $f_s(t)=t^{-\alpha}$ where $\alpha$ is the critical exponent we derive here. For $t>0$ we define the value $r$ as $r=\lim_{t\rightarrow 0}\frac{d\ln{f(t)}}{d\ln{t}}$ %% In the actual notes it says the function for critical exponents are "well behaved" as well as regular functions. What does Well behaved mean?%% where for $t<0$ $r=\lim_{t\rightarrow 0}\frac{d\ln{f_s(t)+f_r(t)}}{d\ln{t}} = -\alpha+\lim_{t\rightarrow 0}\frac{d\ln{1+f_r(t)/f_s(t)}}{d\ln{t}}=-\alpha$ The critical exponent, $\alpha$, may be positive or negative. ## $f$ as a function of [reduced temperature,](reduced%20temperature.md) $t$ for different values of $\alpha.$   ## Renormalization group [[renormalization group]] #StatisticalPhysics