0
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I have simulated a DLA pattern by MC method and the data is for fractal dimension. The right column is the number of particle N(r) into radius r and the left column is the radius r. I plotted a log-log plot by Origin software. But I have a problem with fitting onto the plot. I do not know which part of the data should be considered and others should be deleted to fit on the log-log plot accurately and show me a correct fractal dimension. Please guide me to do it. In fact, I should consider a scaling region for calculating the slope (fractal dimension) but I do not know how I do it.

The result is below:

Graph

Two circles show two regions which should be considered accurately.

My data is shown below:

      r N(r)
      0 1
      1 8
      2 17
      3 24
      4 35
      5 47
      6 61
      7 75
      8 95
      9 116
      10    136
      11    156
      12    175
      13    204
      14    236
      15    265
      16    289
      17    312
      18    343
      19    369
      20    392
      21    418
      22    450
      23    479
      24    502
      25    536
      26    575
      27    609
      28    646
      29    684
      30    721
      31    757
      32    794
      33    837
      34    883
      35    923
      36    978
      37    1018
      38    1054
      39    1107
      40    1151
      41    1204
      42    1247
      43    1303
      44    1360
      45    1416
      46    1476
      47    1523
      48    1573
      49    1621
      50    1666
      51    1715
      52    1749
      53    1785
      54    1831
      55    1896
      56    1950
      57    2019
      58    2071
      59    2130
      60    2177
      61    2236
      62    2304
      63    2349
      64    2403
      65    2477
      66    2542
      67    2610
      68    2680
      69    2738
      70    2811
      71    2882
      72    2946
      73    3027
      74    3111
      75    3198
      76    3288
      77    3384
      78    3474
      79    3573
      80    3660
      81    3732
      82    3818
      83    3886
      84    3979
      85    4061
      86    4129
      87    4211
      88    4284
      89    4385
      90    4470
      91    4538
      92    4614
      93    4689
      94    4762
      95    4832
      96    4921
      97    5007
      98    5097
      99    5193
      100   5284
      101   5369
      102   5470
      103   5565
      104   5660
      105   5755
      106   5853
      107   5959
      108   6052
      109   6159
      110   6263
      111   6376
      112   6503
      113   6611
      114   6719
      115   6809
      116   6893
      117   7001
      118   7112
      119   7225
      120   7335
      121   7434
      122   7580
      123   7734
      124   7868
      125   7991
      126   8117
      127   8245
      128   8383
      129   8520
      130   8670
      131   8785
      132   8914
      133   9041
      134   9171
      135   9297
      136   9429
      137   9557
      138   9675
      139   9803
      140   9929
      141   10037
      142   10143
      143   10274
      144   10429
      145   10549
      146   10659
      147   10766
      148   10873
      149   10978
      150   11086
      151   11196
      152   11315
      153   11445
      154   11559
      155   11681
      156   11815
      157   11947
      158   12081
      159   12218
      160   12370
      161   12536
      162   12669
      163   12812
      164   12941
      165   13061
      166   13176
      167   13307
      168   13433
      169   13566
      170   13689
      171   13812
      172   13935
      173   14052
      174   14167
      175   14256
      176   14348
      177   14453
      178   14561
      179   14676
      180   14768
      181   14840
      182   14915
      183   14995
      184   15065
      185   15139
      186   15195
      187   15253
      188   15325
      189   15404
      190   15489
      191   15570
      192   15649
      193   15745
      194   15835
      195   15940
      196   16038
      197   16146
      198   16248
      199   16356
      200   16483
      201   16570
      202   16650
      203   16747
      204   16847
      205   16955
      206   17046
      207   17152
      208   17252
      209   17353
      210   17441
      211   17542
      212   17627
      213   17725
      214   17804
      215   17893
      216   17983
      217   18079
      218   18161
      219   18240
      220   18343
      221   18440
      222   18546
      223   18644
      224   18729
      225   18821
      226   18921
      227   19008
      228   19090
      229   19179
      230   19260
      231   19328
      232   19404
      233   19491
      234   19575
      235   19647
      236   19728
      237   19786
      238   19861
      239   19939
      240   20016
      241   20089
      242   20161
      243   20233
      244   20295
      245   20378
      246   20450
      247   20515
      248   20619
      249   20692
      250   20780
      251   20846
      252   20925
      253   20982
      254   21041
      255   21102
      256   21151
      257   21206
      258   21263
      259   21334
      260   21383
      261   21434
      262   21498
      263   21570
      264   21670
      265   21769
      266   21846
      267   21907
      268   21973
      269   22047
      270   22113
      271   22178
      272   22232
      273   22278
      274   22307
      275   22339
      276   22387
      277   22432
      278   22497
      279   22555
      280   22616
      281   22680
      282   22728
      283   22775
      284   22822
      285   22856
      286   22886
      287   22925
      288   22961
      289   23004
      290   23045
      291   23093
      292   23128
      293   23185
      294   23239
      295   23283
      296   23330
      297   23373
      298   23415
      299   23452
      300   23481
      301   23528
      302   23576
      303   23627
      304   23671
      305   23705
      306   23735
      307   23771
      308   23825
      309   23871
      310   23910
      311   23954
      312   23994
      313   24033
      314   24066
      315   24093
      316   24126
      317   24156
      318   24201
      319   24227
      320   24249
      321   24277
      322   24290
      323   24309
      324   24357
      325   24402
      326   24450
      327   24483
      328   24514
      329   24536
      330   24574
      331   24605
      332   24625
      333   24655
      334   24684
      335   24701
      336   24725
      337   24747
      338   24761
      339   24784
      340   24809
      341   24830
      342   24852
      343   24870
      344   24891
      345   24906
      346   24917
      347   24931
      348   24945
      349   24959
      350   24971
      351   24982
      352   24990
      353   24993
      354   24995
      355   24999
      356   25000
      357   25001
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$\endgroup$
  • 2
    $\begingroup$ What does the data denote? There are different ways of calculating fractal dimensions (box counting seems likely here). Also, how do you know if you have the "correct" fractal dimension? Please refine your question and show what you are trying to do. $\endgroup$ – Anders Sandberg Jun 6 at 22:51
  • $\begingroup$ It is often (though not absolutely always) the case that the fitting problem is formally independent of the physics. That is, getting fit parameters out of a log-log plot is a problem that comes up over and over again and the way you do it has nothing to do with the fact that the data arose int he context of trying to learn something about fractal dimension. $\endgroup$ – dmckee Jun 7 at 23:04
  • $\begingroup$ @AndersSandberg Thank you for your response. Actually, I have simulated a DLA pattern by MC method and the data is for fractal dimension. The left column is the number of particle N(r) into radius r and the right column is the radius r. I plotted a log-log plot by Origin software. But I have a problem with fitting onto the plot. In fact, I should consider a scaling region for calculating the slope (fractal dimension) but I do not know how I do it. $\endgroup$ – Alireza Alipour Jun 8 at 6:55
0
$\begingroup$

The value you plotted onto the fit is the one you are interested in.

Note that you are computing a quantity different from the box-dimension, which is the quantity generally used for these estimates.

The box dimension is defined as:

$$ d_B=lim_{\epsilon\rightarrow0}\frac{log\;N(\epsilon)}{log\;(\frac{1}{\epsilon})} $$ While you are using the quantity: $d=\frac{log\;N(R)}{log\;R}$. If you substitute $\epsilon$ with $R^{-1}$ in the definition of box dimension then, in the limit of $R\rightarrow\infty$ you obtain the same value, so that you can interchange the two estimates whenever $R$ is large enough, say so large that only a negligible set of your particles are distant less than $R^{-1}$.

This said, let's come to the plot.

On the rightmost part, is obvious that you are considering values of $R$ for which you are encircling almost all of your particles, so you should discard your data.

On the leftmost side, on the other hand, you have a sort of offset in your dimension which is most likely due to making your estimate on a local cluster of particles. At this stage this part of the data is not to be considered either.

In order to understand which data to discard on the left side you should:

  • check that the particles are almost all at distances larger than $R_{min}^{-1}$ from each others;

  • average the estimate around your distribution of particles for the smaller values of the radius;

This last step will give you some error bars to put on the data and probably will also reduce the offset you are seeing on the left side of your graph. Be careful to implement this averaging without including (almost empty) balls on the edge of your distribution, because these would ruin your estimate.


Edit: looking at the numbers I realize that probably most of your data are not in the range that makes your quantity equivalent to the box dimension. However I would say that the quantity you are considering may have a dignity of its own as scaling dimension. I don't know how much this can be similar to other estimates of fractal dimension, like the box dimension or the Haussdorff dimension.

You could make a separate fit for the region in which $R$ is so large that (almost) all the particles are at distances larger than $R^{-1}$ and see the difference between the results.

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  • $\begingroup$ Thank you so much for your response. I have made a mistake on my data and corrected it. In fact, The right column is the number of particle N(r) into radius r and the left column is the radius r. But about your answer, I should be mention that the main answer for solving the problem is to find R min and R max. After that, we can fit over the plot in the scaling region (R min < R < R max). Is there a way to find the region from the data? (size of particles equals to lattice constant (unit) and size of the square lattice is 1024 * 1024) $\endgroup$ – Alireza Alipour Jun 9 at 8:28
  • $\begingroup$ yeah basically when $N_{tot} - N(R) < N(R) - N(R_{previous}) $ you approximately have reached $R_{max} $ (where the subscript previous indicates the R used for the before last estimate), while when $\sqrt{\pi R^2/N(R)}>1/R $ you are at sufficiently large values of $R$ (the left hand side is a very rough estimate of the distance between the particles). To have the correct estimate of you R min you should compute the distribution of the distance between nearest particles and use 1/R< dmin, where you take dmin to be a distance so small that say 95% of your particles have their nearest neighbour $\endgroup$ – AoZora Jun 9 at 13:32
  • $\begingroup$ * nearest neighbour at a larger distance than dmin. Sorry for the format but I can't do better right now $\endgroup$ – AoZora Jun 9 at 13:35
  • $\begingroup$ Thank you so much for your answers. I want to find R min and R max on above data and based on your explanation, actually, I did not understand to find it. Would you mind helping me to find an exact number for R min and R max on above data? $\endgroup$ – Alireza Alipour Jun 10 at 6:28
  • $\begingroup$ sorry but I have no time to do that. Let me explain a bit more precisely two things: basically the $R_{max}$ will be where the (approximatively) linear growth of $log\;N(R)$ stops, the formula I gave you is just a way to extract that computationally. To find the $R_{min}$ you can first use the inequality with pi and square root, that is a good approximation if your distribution of particles does not have too many inhomogeneities. If it does you need to estimate the shortest distance of separation between the particles. $\endgroup$ – AoZora Jun 10 at 6:50

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