\documentclass[12pt]{article}

\begin{document}

$$

\mathbf{J}  

= \sum_i e_i \mathbf{v}_i

+ \sum_{i<j} \left( \mathbf{f}_{ij} \cdot \mathbf{v}_j \right) \mathbf{x}_{ij}

= \sum_i e_i \mathbf{v}_i

+ \frac{1}{2} \sum_{i<j} \left( \mathbf{f}_{ij} \cdot \left(\mathbf{v}_i + \mathbf{v}_j \right)  \right) \mathbf{x}_{ij}

$$

\end{document}

\documentclass[12pt]{article}

\begin{document}

$$

\kappa  = \frac{V}{k_B T^2} \int_0^\infty \langle J_x(0)  J_x(t) \rangle \, dt

= \frac{V}{3 k_B T^2} \int_0^\infty \langle \mathbf{J}(0) \cdot  \mathbf{J}(t)  \rangle \, dt

$$

\end{document}

#!/usr/bin/python

"""

  function: 

    parse the block of thermo data in a lammps logfile and perform auto- and 

    cross correlation of the specified column data.  The total sum of the 

    correlation is also computed which can be converted to an integral by 

    multiplying by the timestep. 

  output:

    standard output contains column data for the auto- & cross correlations 

    plus the total sum of each. Note, only the upper triangle of the 

    correlation matrix is computed.

  usage: 

    correlate.py [-c col] <-c col2> <-s max_correlation_time>  [logfile] 

"""

import sys

import re

import array

# parse command line

maxCorrelationTime = 0

cols = array.array("I")

nCols = 0

args = sys.argv[1:]

index = 0

while index < len(args):

  arg = args[index]

  index += 1

  if   (arg == "-c"):

    cols.append(int(args[index])-1)

    nCols += 1

    index += 1

  elif (arg == "-s"):

    maxCorrelationTime = int(args[index])

    index += 1

  else :

    filename = arg

if (nCols < 1): raise RuntimeError, 'no data columns requested'

data = [array.array("d")]

for s in range(1,nCols)  : data.append( array.array("d") )

# read data block from log file

start = False

input = open(filename)

nSamples = 0

pattern = re.compile('\d')

line = input.readline()

while line :

  columns = line.split()

  if (columns and pattern.match(columns[0])) :

    for i in range(nCols): 

      data[i].append( float(columns[cols[i]]) )

    nSamples += 1

    start = True

  else :

     if (start) : break

  line = input.readline()

print "# read :",nSamples," samples of ", nCols," data"

if( maxCorrelationTime < 1): maxCorrelationTime = int(nSamples/2);

# correlate and integrate

correlationPairs = []

for i in range(0,nCols):

  for j in range(i,nCols): # note only upper triangle of the correlation matrix

    correlationPairs.append([i,j])

header = "# "

for k in range(len(correlationPairs)):

  i = str(correlationPairs[k][0]+1)

  j = str(correlationPairs[k][1]+1)

  header += " C"+i+j+" sum_C"+i+j

print header

nCorrelationPairs = len(correlationPairs)

sum = [0.0] * nCorrelationPairs

for s in range(maxCorrelationTime)  :

  correlation = [0.0] * nCorrelationPairs

  nt = nSamples-s

  for t in range(0,nt)  :

    for p in range(nCorrelationPairs):

      i = correlationPairs[p][0]

      j = correlationPairs[p][1]

      correlation[p] += data[i][t]*data[j][s+t]

  output = ""

  for p in range(0,nCorrelationPairs):

    correlation[p] /= nt

    sum[p] += correlation[p]

    output += str(correlation[p]) + " " + str(sum[p]) + " "

  print output