<description/> <author>Andrei Tokmakoff</author> <company/> <keywords/> <revisedBy>Andrei Tokmakoff</revisedBy> </userData> <identityInfo> <revision>9</revision> <documentID>69253B96-5376-44E7-A25B-784BF8E8C518</documentID> <versionID>F39A5A87-0C3F-4A9B-92F5-4819EFAA9857</versionID> <parentVersionID>00000000-0000-0000-0000-000000000000</parentVersionID> <branchID>00000000-0000-0000-0000-000000000000</branchID> </identityInfo> </metadata> <settings> <presentation> <textRendering> <textStyles> <textStyle name="Normal"> <blockAttr margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Heading 1"> <blockAttr margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"/> 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<units> <currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="true" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="423" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="208" left="18" top="17.25" width="421.5" height="15.75" align-x="18" align-y="30" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f size="14">A worksheet for generating trajectories for a Gaussian random process</f> </text> </region> <region region-id="333" left="18" top="38.25" width="421.5" height="84" align-x="18" align-y="48" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> This worksheet gnerates a trajectory of points for a Gaussian random process characterized by an arbitrary time-correlation function.<sp count="2"/>The example specifically calculates a trajectory of instantaneous frequencies for a quantum system coupled to a bath. The method makes use of the convolution theorem.<sp count="2"/>The convolution of a set of Gaussian random time points with a time-correlation function can be evaluated by taking the product of their Fourier transforms and inverse Fourier transforming.<sp count="3"/> </text> </region> <region region-id="332" left="18" top="140.25" width="164.25" height="12" align-x="18" align-y="150" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> The number of points in the trajectory: </text> </region> <region region-id="325" left="210" top="135" width="34.5" height="18.75" align-x="222.75" align-y="150" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">T</ml:id> <ml:apply> <ml:pow/> <ml:real>2</ml:real> <ml:real>14</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="324" left="294" top="135" width="69.75" height="18.75" align-x="305.25" align-y="150" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">T</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>16384</ml:real> </result> </ml:eval> </math> <rendering item-idref="3"/> </region> <region region-id="235" left="492" top="146.25" width="67.5" height="27.75" align-x="503.25" align-y="162" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math error="This variable is undefined." optimize="false" 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<region region-id="237" left="96" top="183" width="36" height="18" align-x="111" align-y="192" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">τ</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">δt</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="319" left="534" top="195" width="45" height="12.75" align-x="557.25" align-y="204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">tim2</ml:id> <ml:real>150</ml:real> </ml:define> </math> <rendering item-idref="11"/> </region> <region region-id="320" left="600" top="195" width="45" height="12.75" align-x="623.25" align-y="204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIFunction" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">time</ml:id> <ml:real>590</ml:real> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="361" left="18" top="212.25" width="153.75" height="12" align-x="30.75" align-y="222" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Define time-correlation function: </text> </region> <region region-id="390" left="54" top="237" width="27" height="12.75" align-x="68.25" align-y="246" show-border="false" show-highlight="false" 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align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Fill data array with TCF: </text> </region> <region region-id="268" left="174" top="261" width="45" height="18" align-x="189" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">E</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">τ</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="18"/> </region> <region region-id="362" left="18" top="290.25" width="136.5" height="12" align-x="30.75" align-y="300" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Define Gaussian distribution: </text> </region> <region region-id="386" left="54" top="314.25" width="431.25" height="27" align-x="71.25" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Fill an array with a set of Gaussian distributed random numbers. Centered about ω<f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> 0 </inlineAttr> </f>=0 and width σ. </text> </region> <region region-id="398" left="60" top="351" width="34.5" height="12.75" align-x="72.75" align-y="360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">σ</ml:id> <ml:real>180</ml:real> </ml:define> </math> <rendering item-idref="19"/> </region> <region region-id="399" left="102" top="347.25" width="20.25" height="15" align-x="102" align-y="360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> cm<f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> -1 </inlineAttr> </f> </text> </region> <region region-id="400" left="162" top="351" width="77.25" height="12.75" align-x="174" align-y="360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">P</ml:id> <ml:apply> <ml:id xml:space="preserve">rnorm</ml:id> <ml:sequence> <ml:id xml:space="preserve">T</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">σ</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="363" left="18" top="374.25" width="297" height="24" align-x="18" align-y="384" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Convolute the random noise with the exponential correlation function. Do in the frequency domain with FT. </text> </region> <region region-id="354" left="60" top="411" width="49.5" height="12.75" align-x="78.75" align-y="420" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Pω</ml:id> <ml:apply> <ml:id xml:space="preserve">fft</ml:id> <ml:id xml:space="preserve">P</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="301" left="126" top="411" width="51" height="12.75" align-x="145.5" align-y="420" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Eω</ml:id> <ml:apply> <ml:id xml:space="preserve">fft</ml:id> <ml:id 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push-down="false" lock-width="true"> You may want to correct the average of the data set for imperfect sampling: </text> </region> <region region-id="353" left="60" top="458.25" width="105" height="12" align-x="60" align-y="468" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Frequency Fluctuations: </text> </region> <region region-id="313" left="180" top="453" width="110.25" height="18.75" align-x="196.5" align-y="468" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">δP</ml:id> <ml:apply> <ml:vectorize/> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">Re</ml:id> <ml:id xml:space="preserve">P2</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">mean</ml:id> <ml:id xml:space="preserve">P2</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="365" left="18" top="482.25" width="186.75" height="12" align-x="18" align-y="492" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Add back a mean value for the fluctuations: </text> </region> <region region-id="406" left="60" top="507" width="43.5" height="15" align-x="77.25" align-y="516" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="0">ω</ml:id> <ml:real>1804</ml:real> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="407" left="114" top="503.25" width="20.25" height="15" align-x="114" align-y="516" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> cm<f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> -1 </inlineAttr> </f> </text> </region> <region region-id="408" left="156" top="501" width="63" height="21" align-x="171" align-y="516" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Pt</ml:id> <ml:apply> <ml:vectorize/> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="0">ω</ml:id> <ml:id xml:space="preserve">δP</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="26"/> </region> <region region-id="409" left="18" top="522" width="471.75" height="147.75" align-x="18" align-y="522" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="27"/> <rendering item-idref="28"/> </region> <region region-id="405" left="0" top="672" width="6000" height="6" align-x="0" align-y="672" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <pageBreak/> </region> <region region-id="410" left="24" top="686.25" width="68.25" height="12" align-x="24" align-y="696" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Export data file: </text> </region> <region region-id="413" left="24" top="705" width="122.25" height="12.75" align-x="130.5" align-y="714" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:unknownOp/> <ml:apply> <ml:id xml:space="preserve">WRITEPRN</ml:id> <ml:str xml:space="preserve">D:\cf.dat</ml:str> </ml:apply> <ml:id xml:space="preserve">Pt</ml:id> </ml:apply> <resultFormat numeric-only="true"> <general precision="4" show-trailing-zeros="true" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="8"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="29"/> </region> <region region-id="415" left="24" top="740.25" width="426" height="51" align-x="42" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Calculate a correlation function from the trajectory: <tab/>We'll evaluate products of pairs of frequency separated by t, and normalize by the number <tab/>of observations.<sp count="2"/>Since we are interested in the short time behavior, let's not include all of <tab/>the products for time delays >> τ<f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> c </inlineAttr> </f>. </text> </region> <region region-id="49" left="462" top="758.25" width="133.5" height="12" align-x="462" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Frequency correlation function: </text> </region> <region region-id="417" left="60" top="801" width="45.75" height="12.75" align-x="72" align-y="810" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">q</ml:id> <ml:range> <ml:real>0</ml:real> <ml:real>600</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="418" left="90" top="829.5" width="144.75" height="43.5" align-x="110.25" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Ct</ml:id> <ml:id xml:space="preserve">q</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">T</ml:id> <ml:id xml:space="preserve">q</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">δP</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">q</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">δP</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">T</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:id xml:space="preserve">q</ml:id> </ml:apply> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="31"/> </region> <region region-id="422" left="54" top="912" width="385.5" height="196.5" align-x="54" align-y="912" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> 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<math error="This value must be a scalar." optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">iii</ml:id> <ml:range> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">T</ml:id> <ml:id xml:space="preserve">time</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="238" left="24" top="183" width="51.75" height="12.75" align-x="33" align-y="192" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">i</ml:id> <ml:range> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">T</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="237" left="96" top="183" width="36" height="18" align-x="111" align-y="192" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">τ</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">δt</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="319" left="534" top="195" width="45" height="12.75" align-x="557.25" align-y="204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">tim2</ml:id> <ml:real>150</ml:real> </ml:define> </math> <rendering item-idref="11"/> </region> <region region-id="320" left="600" top="195" width="45" height="12.75" align-x="623.25" align-y="204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIFunction" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">time</ml:id> <ml:real>590</ml:real> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="361" left="18" top="212.25" width="153.75" height="12" align-x="30.75" align-y="222" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Define time-correlation function: </text> </region> <region region-id="390" left="54" top="237" width="27" height="12.75" align-x="68.25" align-y="246" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">A</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="389" left="96" top="237" width="36.75" height="15" align-x="111" align-y="246" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <ml:real>200</ml:real> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="388" left="162" top="230.25" width="77.25" height="30" align-x="185.25" align-y="246" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">C</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="c">τ</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="397" left="294" top="246" width="206.25" height="137.25" align-x="294" align-y="246" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="16"/> <rendering item-idref="17"/> </region> <region region-id="267" left="54" top="260.25" width="103.5" height="12" align-x="60" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Fill data array with TCF: </text> </region> <region region-id="268" left="174" top="261" width="45" height="18" align-x="189" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">E</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">τ</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="18"/> </region> <region region-id="362" left="18" top="290.25" width="136.5" height="12" align-x="30.75" align-y="300" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Define Gaussian distribution: </text> </region> <region region-id="386" left="54" top="314.25" width="431.25" height="27" align-x="71.25" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Fill an array with a set of Gaussian distributed random numbers. Centered about ω<f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> 0 </inlineAttr> </f>=0 and width σ. </text> </region> <region region-id="398" left="60" top="351" width="34.5" height="12.75" align-x="72.75" align-y="360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">σ</ml:id> <ml:real>180</ml:real> </ml:define> </math> <rendering item-idref="19"/> </region> <region region-id="399" left="102" top="347.25" width="20.25" height="15" align-x="102" align-y="360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> cm<f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> -1 </inlineAttr> </f> </text> </region> <region region-id="400" left="162" top="351" width="77.25" height="12.75" align-x="174" align-y="360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">P</ml:id> <ml:apply> <ml:id xml:space="preserve">rnorm</ml:id> <ml:sequence> <ml:id xml:space="preserve">T</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">σ</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="363" left="18" top="374.25" width="297" height="24" align-x="18" align-y="384" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Convolute the random noise with the exponential correlation function. Do in the frequency domain with FT. </text> </region> <region region-id="354" left="60" top="411" width="49.5" height="12.75" align-x="78.75" align-y="420" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Pω</ml:id> <ml:apply> <ml:id xml:space="preserve">fft</ml:id> <ml:id xml:space="preserve">P</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="301" left="126" top="411" width="51" height="12.75" align-x="145.5" align-y="420" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Eω</ml:id> <ml:apply> <ml:id xml:space="preserve">fft</ml:id> <ml:id xml:space="preserve">E</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="22"/> </region> <region region-id="299" left="204" top="405" width="78.75" height="18.75" align-x="220.5" align-y="420" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">P2</ml:id> <ml:apply> <ml:id xml:space="preserve">ifft</ml:id> <ml:apply> <ml:vectorize/> <ml:parens> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">Pω</ml:id> <ml:id xml:space="preserve">Eω</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="364" left="18" top="434.25" width="325.5" height="12" align-x="27" align-y="444" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> You may want to correct the average of the data set for imperfect sampling: </text> </region> <region region-id="353" left="60" top="458.25" width="105" height="12" align-x="60" align-y="468" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Frequency Fluctuations: </text> </region> <region region-id="313" left="180" top="453" width="110.25" height="18.75" align-x="196.5" align-y="468" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">δP</ml:id> <ml:apply> <ml:vectorize/> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">Re</ml:id> <ml:id xml:space="preserve">P2</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">mean</ml:id> <ml:id xml:space="preserve">P2</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="365" left="18" top="482.25" width="186.75" height="12" align-x="18" align-y="492" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Add back a mean value for the fluctuations: </text> </region> <region region-id="406" left="60" top="507" width="43.5" height="15" align-x="77.25" align-y="516" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="0">ω</ml:id> <ml:real>1804</ml:real> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="407" left="114" top="503.25" width="20.25" height="15" align-x="114" align-y="516" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> cm<f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> -1 </inlineAttr> </f> </text> </region> <region region-id="408" left="156" top="501" width="63" height="21" align-x="171" align-y="516" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Pt</ml:id> <ml:apply> <ml:vectorize/> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="0">ω</ml:id> <ml:id xml:space="preserve">δP</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="26"/> </region> <region region-id="409" left="18" top="522" width="471.75" height="147.75" align-x="18" align-y="522" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="27"/> <rendering item-idref="28"/> </region> <region region-id="405" left="0" top="672" width="6000" height="6" align-x="0" align-y="672" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <pageBreak/> </region> <region region-id="410" left="24" top="686.25" width="68.25" height="12" align-x="24" align-y="696" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Export data file: </text> </region> <region region-id="413" left="24" top="705" width="122.25" height="12.75" align-x="130.5" align-y="714" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:unknownOp/> <ml:apply> <ml:id xml:space="preserve">WRITEPRN</ml:id> <ml:str xml:space="preserve">D:\cf.dat</ml:str> </ml:apply> <ml:id xml:space="preserve">Pt</ml:id> </ml:apply> <resultFormat numeric-only="true"> <general precision="4" show-trailing-zeros="true" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="8"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="29"/> </region> <region region-id="415" left="24" top="740.25" width="426" height="51" align-x="42" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Calculate a correlation function from the trajectory: <tab/>We'll evaluate products of pairs of frequency separated by t, and normalize by the number <tab/>of observations.<sp count="2"/>Since we are interested in the short time behavior, let's not include all of <tab/>the products for time delays >> τ<f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> c </inlineAttr> </f>. </text> </region> <region region-id="49" left="462" top="758.25" width="133.5" height="12" align-x="462" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Frequency correlation function: </text> </region> <region region-id="417" left="60" top="801" width="45.75" height="12.75" align-x="72" align-y="810" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">q</ml:id> <ml:range> <ml:real>0</ml:real> <ml:real>600</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="418" left="90" top="829.5" width="144.75" height="43.5" align-x="110.25" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Ct</ml:id> <ml:id xml:space="preserve">q</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">T</ml:id> <ml:id xml:space="preserve">q</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">δP</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">q</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">δP</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">T</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:id xml:space="preserve">q</ml:id> </ml:apply> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="31"/> </region> <region region-id="422" left="54" top="912" width="385.5" height="196.5" align-x="54" align-y="912" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> 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