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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Linear response function</span></span>

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<p>A <b>linear response function</b> describes the input-output relationship of a <a href="Signal_transducer" class="mw-redirect" title="Signal transducer">signal transducer</a>, such as a radio turning <a href="Electromagnetic_wave" class="mw-redirect" title="Electromagnetic wave">electromagnetic waves</a> into music or a <a href="Neuron" title="Neuron">neuron</a> turning <a href="Synapse" title="Synapse">synaptic</a> input into a response. Because of its many applications in <a href="Information_theory" title="Information theory">information theory</a>, <a href="Physics" title="Physics">physics</a> and <a href="Engineering" title="Engineering">engineering</a> there exist alternative names for specific linear response functions such as <a href="Magnetic_susceptibility" title="Magnetic susceptibility">susceptibility</a>, <a href="Impulse_response" title="Impulse response">impulse response</a> or <a href="Electrical_impedance" title="Electrical impedance">impedance</a>; see also <a href="Transfer_function" title="Transfer function">transfer function</a>. The concept of a <a href="Green's_function" title="Green's function">Green's function</a> or <a href="Fundamental_solution" title="Fundamental solution">fundamental solution</a> of an <a href="Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a> is closely related.

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<div class="mw-heading mw-heading2"><h2 id="Mathematical_definition">Mathematical definition</h2></div>

<p>Denote the input of a system by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(t)}">

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<annotation encoding="application/x-tex">{\displaystyle h(t)}</annotation>

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</math></span><img src="./66abbb8ae1d9f30bb529739b109e1e5bbe83c626.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.988ex; height:2.843ex;" alt="{\displaystyle h(t)}" loading="lazy"></span> (e.g. a <a href="Force" title="Force">force</a>), and the response of the system by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

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<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

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</math></span><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"></span> (e.g. a position). Generally, the value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

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<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

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</math></span><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"></span> will depend not only on the present value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(t)}">

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<annotation encoding="application/x-tex">{\displaystyle h(t)}</annotation>

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</math></span><img src="./66abbb8ae1d9f30bb529739b109e1e5bbe83c626.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.988ex; height:2.843ex;" alt="{\displaystyle h(t)}" loading="lazy"></span>, but also on past values. Approximately <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

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<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

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</math></span><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"></span> is a weighted sum of the previous values of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(t')}">

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<annotation encoding="application/x-tex">{\displaystyle h(t')}</annotation>

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</math></span><img src="./97ba61f7631d05e567722ceed2dbdb95cfa7efab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.673ex; height:3.009ex;" alt="{\displaystyle h(t')}" loading="lazy"></span>, with the weights given by the linear response function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi (t-t')}">

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<annotation encoding="application/x-tex">{\displaystyle \chi (t-t')}</annotation>

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</math></span><img src="./a6f323a000ba767e8f16d96b0cef08e09d82c746.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.469ex; height:3.009ex;" alt="{\displaystyle \chi (t-t')}" loading="lazy"></span>:

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<annotation encoding="application/x-tex">{\displaystyle x(t)=\int _{-\infty }^{t}dt'\,\chi (t-t')h(t')+\cdots \,.}</annotation>

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</p><p>The explicit term on the right-hand side is the <a href="Leading-order" class="mw-redirect" title="Leading-order">leading order</a> term of a <a href="Volterra_series" title="Volterra series">Volterra expansion</a> for the full nonlinear response. If the system in question is highly non-linear, higher order terms in the expansion, denoted by the dots, become important and the signal transducer cannot adequately be described just by its linear response function.

</p><p>The complex-valued <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\chi }}(\omega )}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {\chi }}(\omega )}</annotation>

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</math></span><img src="./fce561dfac738a9f5d80162d0eceb29b8f91b80a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.71ex; height:2.843ex;" alt="{\displaystyle {\tilde {\chi }}(\omega )}" loading="lazy"></span> of the linear response function is very useful as it describes the output of the system if the input is a sine wave <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(t)=h_{0}\sin(\omega t)}">

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<annotation encoding="application/x-tex">{\displaystyle h(t)=h_{0}\sin(\omega t)}</annotation>

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</math></span><img src="./3bdf4dd7cb5fccb233ca4593c55d59ae71e7b3f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.817ex; height:2.843ex;" alt="{\displaystyle h(t)=h_{0}\sin(\omega t)}" loading="lazy"></span> with frequency <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

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</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span>. The output reads

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)=\left|{\tilde {\chi }}(\omega )\right|h_{0}\sin(\omega t+\arg {\tilde {\chi }}(\omega ))\,,}">

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<annotation encoding="application/x-tex">{\displaystyle x(t)=\left|{\tilde {\chi }}(\omega )\right|h_{0}\sin(\omega t+\arg {\tilde {\chi }}(\omega ))\,,}</annotation>

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</p><p>with <a href="Amplifier" title="Amplifier">amplitude gain</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\tilde {\chi }}(\omega )|}">

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<annotation encoding="application/x-tex">{\displaystyle |{\tilde {\chi }}(\omega )|}</annotation>

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</math></span><img src="./0ab95c4987b9c21af38bb971e30466128368fd64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.004ex; height:2.843ex;" alt="{\displaystyle |{\tilde {\chi }}(\omega )|}" loading="lazy"></span> and <a href="Phase_shift" class="mw-redirect" title="Phase shift">phase shift</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arg {\tilde {\chi }}(\omega )}">

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<annotation encoding="application/x-tex">{\displaystyle \arg {\tilde {\chi }}(\omega )}</annotation>

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</math></span><img src="./88352731ff73714e20f75fa21eba84c0722e9158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.334ex; height:2.843ex;" alt="{\displaystyle \arg {\tilde {\chi }}(\omega )}" loading="lazy"></span>.

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<div class="mw-heading mw-heading2"><h2 id="Example">Example</h2></div>

<p>Consider a <a href="Damped_harmonic_oscillator" class="mw-redirect" title="Damped harmonic oscillator">damped harmonic oscillator</a> with input given by an external driving force <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(t)}">

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<annotation encoding="application/x-tex">{\displaystyle h(t)}</annotation>

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</math></span><img src="./66abbb8ae1d9f30bb529739b109e1e5bbe83c626.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.988ex; height:2.843ex;" alt="{\displaystyle h(t)}" loading="lazy"></span>,

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {x}}(t)+\gamma {\dot {x}}(t)+\omega _{0}^{2}x(t)=h(t).}">

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<annotation encoding="application/x-tex">{\displaystyle {\ddot {x}}(t)+\gamma {\dot {x}}(t)+\omega _{0}^{2}x(t)=h(t).}</annotation>

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</p><p>The complex-valued <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of the linear response function is given by

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\chi }}(\omega )={\frac {{\tilde {x}}(\omega )}{{\tilde {h}}(\omega )}}={\frac {1}{\omega _{0}^{2}-\omega ^{2}+i\gamma \omega }}.}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {\chi }}(\omega )={\frac {{\tilde {x}}(\omega )}{{\tilde {h}}(\omega )}}={\frac {1}{\omega _{0}^{2}-\omega ^{2}+i\gamma \omega }}.}</annotation>

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</p><p>The amplitude gain is given by the magnitude of the <a href="Complex_number" title="Complex number">complex number</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\chi }}(\omega ),}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {\chi }}(\omega ),}</annotation>

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</math></span><img src="./badfaa21b5a9adb1977f4fc0063893bd109ddf02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.357ex; height:2.843ex;" alt="{\displaystyle {\tilde {\chi }}(\omega ),}" loading="lazy"></span> and the phase shift by the arctan of the imaginary part of the function divided by the real one.

</p><p>From this representation, we see that for small <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

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</math></span><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"></span> the Fourier transform <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\chi }}(\omega )}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {\chi }}(\omega )}</annotation>

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</math></span><img src="./fce561dfac738a9f5d80162d0eceb29b8f91b80a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.71ex; height:2.843ex;" alt="{\displaystyle {\tilde {\chi }}(\omega )}" loading="lazy"></span> of the linear response function yields a pronounced maximum ("<a href="Resonance" title="Resonance">Resonance</a>") at the frequency <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \approx \omega _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega \approx \omega _{0}}</annotation>

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</math></span><img src="./211f33ab655457ff44c4337e3240c52014c63050.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.044ex; height:2.009ex;" alt="{\displaystyle \omega \approx \omega _{0}}" loading="lazy"></span>. The linear response function for a harmonic oscillator is mathematically identical to that of an <a href="RLC_circuit" title="RLC circuit">RLC circuit</a>. The width of the maximum, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \omega ,}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta \omega ,}</annotation>

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</math></span><img src="./bddb09692243b47e0fdaeceed99e20a8cb024389.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.029ex; height:2.509ex;" alt="{\displaystyle \Delta \omega ,}" loading="lazy"></span> typically is much smaller than <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{0},}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{0},}</annotation>

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</math></span><img src="./f191aa60002cbd513aea6b88666c5500aa84c2fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.147ex; height:2.009ex;" alt="{\displaystyle \omega _{0},}" loading="lazy"></span> so that the <a href="Quality_factor" class="mw-redirect" title="Quality factor">Quality factor</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q:=\omega _{0}/\Delta \omega }">

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<annotation encoding="application/x-tex">{\displaystyle Q:=\omega _{0}/\Delta \omega }</annotation>

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</math></span><img src="./b0116ad92d1a8bd104ab5270bc2b87ca03d72a5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.628ex; height:2.843ex;" alt="{\displaystyle Q:=\omega _{0}/\Delta \omega }" loading="lazy"></span> can be extremely large.

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<div class="mw-heading mw-heading2"><h2 id="Kubo_formula">Kubo formula</h2></div>

<p>The exposition of linear response theory, in the context of <a href="Quantum_statistics" class="mw-redirect" title="Quantum statistics">quantum statistics</a>, can be found in a paper by <a href="Ryogo_Kubo" title="Ryogo Kubo">Ryogo Kubo</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> This defines particularly the <a href="Kubo_formula" title="Kubo formula">Kubo formula</a>, which considers the general case that the "force" <span class="texhtml"><i>h</i>(<i>t</i>)</span> is a perturbation of the basic operator of the system, the <a href="Hamiltonian_function" class="mw-redirect" title="Hamiltonian function">Hamiltonian</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}_{0}\to {\hat {H}}_{0}-h(t'){\hat {B}}(t')}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {H}}_{0}\to {\hat {H}}_{0}-h(t'){\hat {B}}(t')}</annotation>

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</math></span><img src="./3f5be9267a141f33b9d1237eafd26b9203957f3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.46ex; height:3.343ex;" alt="{\displaystyle {\hat {H}}_{0}\to {\hat {H}}_{0}-h(t'){\hat {B}}(t')}" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {B}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {B}}}</annotation>

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</math></span><img src="./8ffa3bc40e590b7dfa0cd144e13163abf26e5c38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.843ex;" alt="{\displaystyle {\hat {B}}}" loading="lazy"></span> corresponds to a measurable quantity as input, while the output <span class="texhtml"><i>x</i>(<i>t</i>)</span> is the perturbation of the thermal expectation of another measurable quantity <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {A}}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {A}}(t)}</annotation>

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</math></span><img src="./8162d2ff06f25fa27d2cb08c270eed3a5fe125df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.425ex; height:3.343ex;" alt="{\displaystyle {\hat {A}}(t)}" loading="lazy"></span>. The Kubo formula then defines the quantum-statistical calculation of the <a href="Susceptibility_(disambiguation)" class="mw-redirect mw-disambig" title="Susceptibility (disambiguation)">susceptibility</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi (t-t')}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi (t-t')}</annotation>

</semantics>

</math></span><img src="./a6f323a000ba767e8f16d96b0cef08e09d82c746.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.469ex; height:3.009ex;" alt="{\displaystyle \chi (t-t')}" loading="lazy"></span> by a general formula involving only the mentioned operators.

</p><p>As a consequence of the principle of <a href="Causality_(physics)" title="Causality (physics)">causality</a> the complex-valued function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\chi }}(\omega )}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {\chi }}(\omega )}</annotation>

</semantics>

</math></span><img src="./fce561dfac738a9f5d80162d0eceb29b8f91b80a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.71ex; height:2.843ex;" alt="{\displaystyle {\tilde {\chi }}(\omega )}" loading="lazy"></span> has poles only in the lower half-plane. This leads to the <a href="Kramers%E2%80%93Kronig_relations" title="Kramers–Kronig relations">Kramers–Kronig relations</a>, which relates the real and the imaginary parts of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\chi }}(\omega )}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {\chi }}(\omega )}</annotation>

</semantics>

</math></span><img src="./fce561dfac738a9f5d80162d0eceb29b8f91b80a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.71ex; height:2.843ex;" alt="{\displaystyle {\tilde {\chi }}(\omega )}" loading="lazy"></span> by integration. The simplest example is once more the <a href="Damped_harmonic_oscillator" class="mw-redirect" title="Damped harmonic oscillator">damped harmonic oscillator</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Nonequilibrium_linear_response_formula">Nonequilibrium linear response formula</h2></div>

<p>Linear response theory has versions for nonequilibrium processes for open systems where there is no detailed balance but a steady driving or agitation is applied. A small perturbation of these driven or active systems gives rise to a response in violation with the equilibrium expressions. A possible methodology proceeds via path-space ensembles where the probabilities of trajectories are evaluated; see e.g. <sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The resulting response formulae have an entropic part (similar to the detailed balance case) and a frenetic part. The latter involves the correlation of the (excess) <a href="Frenesy_(physics)" title="Frenesy (physics)">frenesy</a> (due to the perturbation) with the observable. In detailed balance, the two contributions merge and reproduce the Kubo and <a href="Green-Kubo" class="mw-redirect" title="Green-Kubo">Green-Kubo</a> formulae. Out of detailed balance, the frenetic contribution is responsible for the possibility of negative heat capacities and mobilities, and they do not longer measure a fluctuation, e.g. in terms of a variance of the energy or of the current.

</p>

<ul><li><a href="Convolution" title="Convolution">Convolution</a></li>

<li><a href="Green%E2%80%93Kubo_relations" title="Green–Kubo relations">Green–Kubo relations</a></li>

<li><a href="Fluctuation_theorem" title="Fluctuation theorem">Fluctuation theorem</a></li>

<li><a href="Dispersion_(optics)" title="Dispersion (optics)">Dispersion (optics)</a></li>

<li><a href="Lindbladian" title="Lindbladian">Lindbladian</a></li>

<li><a href="Semilinear_response" title="Semilinear response">Semilinear response</a></li>

<li><a href="Green's_function" title="Green's function">Green's function</a></li>

<li><a href="Impulse_response" title="Impulse response">Impulse response</a></li>

<li><a href="Resolvent_formalism" title="Resolvent formalism">Resolvent formalism</a></li>

<li><a href="Propagator" title="Propagator">Propagator</a></li>

<li><a href="Frenesy_(physics)" title="Frenesy (physics)">Frenesy</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Kubo, R., <i>Statistical Mechanical Theory of Irreversible Processes I</i>, Journal of the Physical Society of Japan, vol. <i>12</i>, pp. 570–586 (1957).</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">De Clozeaux,<i>Linear Response Theory</i>, in: E. Antončik et al., <i>Theory of condensed matter</i>, IAEA Vienna, 1968</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Christian Maes, "Frenesy: Time-symmetric dynamical activity in nonequilibria", <i>Physics Reports</i> <b>850</b>, 1–33 (2020).</span>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304030119/http://www.cond-mat.de/events/correl14/manuscripts/pavarini.pdf">Linear Response Functions</a> in Eva Pavarini, Erik Koch, Dieter Vollhardt, and Alexander Lichtenstein (eds.): DMFT at 25: Infinite Dimensions, Verlag des Forschungszentrum Jülich, 2014 <style data-mw-deduplicate="TemplateStyles:r1238218222">

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