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RandomWalk.tm
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<TeXmacs|1.99.12>
<style|<tuple|generic|old-dots>>
<\body>
<doc-data|<doc-title|Note For Random Walk>|<doc-author|<author-data|<author-name|Yuejian
Mo>|<\author-affiliation>
<date|>
</author-affiliation>>>>
<\equation*>
J=-<frac|1|2><frac|N<around*|(|x+\<Delta\>x|)>-N<around*|(|x|)>|A\<tau\>>=-<frac|1|2><frac|<around*|(|C<around*|(|x+\<Delta\>x|)>-C<around*|(|x|)>|)>A\<Delta\>x\<tau\>|A
\<tau\>>=-<frac|\<Delta\>x<rsup|2>|2\<tau\>><frac|C<around*|(|x+\<Delta\>x|)>-C<around*|(|x|)>|\<Delta\>x>=<frac|1|2>D<frac|C<around*|(|x+\<Delta\>x|)>-C<around*|(|x|)>|\<Delta\>x>
</equation*>
\;
<section|Random Walk in 1-D>
In one dimensional random walk of <math|N> particles. A particle always
takes a fixed step size <math|\<Delta\>x> toward the left and right with
equal probability. The position of the particle <math|i> is denoted by
<math|x<rsub|i><around*|(|n|)>>, where <math|n> is the number of steps that
particle <math|i> took. All particles move from <math|x<around*|(|0|)>=0>.\
The position average of <math|N> particles is\
<\equation*>
\<less\>x<around*|(|n|)>\<gtr\>=<big|sum><rsub|i=1><rsup|N>x<rsub|i><around*|(|n|)>=<big|sum><rsub|i=1><rsup|N><around*|(|x<rsub|i><around*|(|n-1|)>\<pm\>\<Delta\>x<rsub|i>|)>=<big|sum><rsub|i=1><rsup|N><around*|(|x<rsub|i><around*|(|0|)>+
<big|sum><rsub|j=0><rsup|n-1>\<pm\>\<Delta\>x<rsub|i>|)>=0
</equation*>
where, I assume that each particle <math|x<rsub|i>> has own
<math|\<Delta\>x<rsub|i>>.
\;
To descript the spread of <math|N> particles, variance is <htab|5mm>
<\equation*>
Var<around*|(|x<around*|(|n|)>|)>=\<less\>x<rsup|2><around*|(|n|)>\<gtr\>-\<less\>x<around*|(|n|)>\<gtr\><rsup|2>=\<less\>x<rsup|2><around*|(|n|)>\<gtr\>=<frac|1|N><big|sum><rsub|i=1><rsup|N>x<rsub|i><rsup|2><around*|(|n|)>=<frac|1|N><big|sum><rsub|i=1><rsup|N><around*|(|x<rsub|i><around*|(|n-1|)>\<pm\>\<Delta\>x<rsub|i>|)><rsup|2>
</equation*>
But here, textbook don't continue expand <math|x<rsub|i><around*|(|n-1|)>>
to <math|x<rsub|i><around*|(|0|)>>, which cause
<math|Var<around*|(|x<around*|(|n|)>|)>=0>. It must has some rules
determine that why we can't do <math|Var<around*|(|x<around*|(|n|)>|)>> as
<math|\<less\>x*<around*|(|n|)>\<gtr\>>. Instead, expand
<math|Var*<around*|(|x<around*|(|n|)>|)>> to
<math|Var<around*|(|x<around*|(|0|)>|)>> \
<\equation*>
Var<around*|(|x<around*|(|n|)>|)>=<frac|1|N><big|sum><rsub|i=1><rsup|N><around*|(|x<rsub|i>*<rsup|2><around*|(|n-1|)><rsup|>\<pm\>2x<rsub|i><around*|(|n-1|)>+\<Delta\>x<rsub|i><rsup|2>|)>=Var<around*|(|x<around*|(|n-1|)>|)>+<frac|1|N><big|sum><rsub|i=1><rsup|N>\<Delta\>x<rsub|i><rsup|2>=<frac|1|N><big|sum><rsub|i=1><rsup|N>n\<Delta\>x<rsub|i><rsup|2>
</equation*>
More, if we assume that all particles move same step size in time
<math|\<tau\>>. During <math|t>, the variance of particles
<math|x<around*|(|n|)>> is
<\equation*>
Var<around*|(|x<around*|(|n|)>|)>=n \<Delta\>x<rsup|2>=t<frac|\<Delta\>x<rsup|2>|\<tau\>>=2
D t
</equation*>
<section|Random Walk under external force>
Fokker-Planck equation describe the time evolution of the p.d.f of the
velocity of a particle under drag forces and random forces, as in Brownian
motion.\
<\equation*>
<frac|\<partial\>|\<partial\>t>f<around*|(|x,t|)>=-<frac|\<partial\>|\<partial\>x><around*|[|D<rsub|1><around*|(|x,t|)>f<around*|(|x,t|)>|]>+<frac|\<partial\><rsup|2>|\<partial\>x<rsup|2>><around*|[|D<rsub|2><around*|(|x,t|)>f<around*|(|x,t|)>|]>
</equation*>
where <math|D<rsub|1><around*|(|x,t|)>> is the drag force parameter,
<math|D<rsub|2><around*|(|x,t|)>> is the diffusion parameter.
<section|Reference>
<hlink|https://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation|https://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation>
</body>
<initial|<\collection>
</collection>>
<\references>
<\collection>
<associate|auto-1|<tuple|1|?>>
<associate|auto-2|<tuple|2|?>>
<associate|auto-3|<tuple|3|?>>
</collection>
</references>
<\auxiliary>
<\collection>
<\associate|toc>
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|1<space|2spc>Random
Walk in 1-D> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-1><vspace|0.5fn>
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|2<space|2spc>Random
Walk under external force> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-2><vspace|0.5fn>
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|3<space|2spc>Reference>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-3><vspace|0.5fn>
</associate>
</collection>
</auxiliary>