Feng Ling PRO
Postdoc @ Helmholtz Pioneer Campus, studying biophysics, cilia, and mucus flow. Looking for Universality from Diversity.
信息热动力学 Part II
Thermodynamics of Information II
报告人：凌峰
亥姆霍兹慕尼黑 / 慕尼黑工业大学 - 亥姆霍兹先锋园区、生物医学成像研究中心
Helmholtz Munich / TUM - Helmholtz Pioneer Campus / IBMI
非平衡统计物理读书会 - 信息热动力学
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包含信息的涨落定理（Fluctuation Theorem）
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涨落定理基本推导简述
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- 稳定记忆与Landauer原理
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测量过程的代价
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连续信息流
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研究前景与展望
- 题外话
- 信息热力学能否用于区分微观环境下的闭环操控与开环涌现？
分享大纲
非平衡统计物理读书会 - 信息热动力学
Recall that without explicitly writing the information component we have
[Clausius Inequality] \(\qquad\qquad\quad\;\; W - \Delta \mathcal F\geq 0\)
[Crooks Fluctuation Theorem] \(\,\quad P(x(t))/P(x_{\text{reverse}}(t)) = \exp[(w-\Delta F)/k_BT]\)
[Integral Fluctuation Theorem] \(\;\;\; \langle e^{-(w-\Delta F)/k_BT}\rangle=1\)
Analogously, from our information-aware Clausius-like inequality (eq.5)
\[W-\Delta \mathcal F\geq -k_BT I\]
One can derive (eq.6)
\[\frac{\mathcal P(\text{forward trajectory},m)}{\mathcal P(\text{reverse trajectory},m)}=e^{(w-\Delta F)/k_BT + \mathcal I}\]
and (eq.7)
\[\langle e^{-(w-\Delta F)/k_BT - \mathcal I} \rangle=1\]
Fluctuation Theorem with Information
非平衡统计物理读书会 - 信息热动力学
Consider a finite, classical system in contact with a thermal bath of temperature \(T\)
let \(\lambda\) denote an externally controllable parameter
\([\lambda,T]\) specifies an equilibrium state
system evolves from equilibrium \([A,T]\) at \(t=0\)
to \(\lambda=B\) at \(t=\tau\) in non-equilibrium,
then relax to equilibrium \([B,T]\) at \(t=\tau*\) (no work performed)
Clausius Inequality \(\leftrightarrow\) Jarzynski Equality
[Clausius Inequality]
\[W=\int_0^\tau\mathrm d t\,\dot{\lambda} \frac{\partial \mathcal H(x(t);\lambda(t))}{\partial \lambda}\geq\Delta F\equiv F_{[B,T]} - F_{[A,T]}\]
Zarzynski (2010), 10.1146/annurev-conmatphys-062910-140506
非平衡统计物理读书会 - 信息热动力学
non-equilibrium
equilibrium bath T
[A,T]
[B,T]
[B]
\(t=0\)
\(t=\tau*\)
\(t=\tau\)
\(W\)
relax
\(\Delta F\)
non-equilibrium work relation \(W = \mathcal H(x_\tau; B) -\mathcal{H} (x_0 ; A)\) gives
\[\langle e^{-W/k_BT}\rangle=\int \mathrm d\bm{x}_0\,p_{[A,T]}(\bm x_0)e^{-W/k_BT}\]
\[=\frac{1}{Z_{[A,T]}}\int \mathrm d\bm{x}_\tau \,e^{-\mathcal H(x_\tau;B)/k_BT}\cancel{\left|\frac{\partial \bm{x}_\tau}{\partial \bm{x}_0}\right|^{-1}}\]
\[=\frac{Z_{[B,T]}}{Z_{[A,T]}}=e^{-\Delta F/k_BT}\]
Clausius Inequality \(\leftrightarrow\) Jarzynski Equality
[Jensen's Inequality] \(\langle exp(\cdot)\rangle\geq exp(\langle\cdot\rangle)\) recovers
\[-\Delta F/k_BT\geq-W/k_BT\text{ or } W\geq\Delta F\]
[Louville's theorem]
Zarzynski (2010), 10.1146/annurev-conmatphys-062910-140506
非平衡统计物理读书会 - 信息热动力学
[Jarzynski Equality]
Note: Jarzynski Equality can be seen as a specific case of the [Integral Fluctuation theorem]
\[\langle e^{-(W-\Delta V)/k_BT} \, \frac{p_1(\bm x_\tau)}{p_0(\bm x_0)}\rangle=1\]
with a specific choice of the arbitrary normalized distribution \(p_1,p_0\) that cancels out the potential term and introduce microstate independent free energy.
Jarzynski Equality provide us a non-equilibirum counterpart to the following equality as a result of reversible work relation \(W=\Delta F\)
\[-k_BT\log\langle e^{-W/k_BT}\rangle=\Delta F\]
Clausius Inequality \(\leftrightarrow\) Jarzynski Equality
Seifert (2008), 10.1140/epjb/e2008-00001-9
Seifert (2005), 10.1103/PhysRevLett.95.040602
非平衡统计物理读书会 - 信息热动力学
非平衡统计物理读书会 - 信息热动力学
The non-equilibrium free energy of memory M is (eq.8)
\[\mathcal F(M) = \langle \mathcal H\rangle_\rho - TS(\rho) = \sum_m p_m F_m - k_BT H(M)\]
Manipulating memory should result in a loop in terms of Hamiltonian.
Thus work to shift one memory state \(p_m\) to another \(p'_m\) satisfies (eq.9)
\[W-\Delta \mathcal F_\text{memory} = W - (\mathcal F(M') - \mathcal F(M))\geq0\]
for symmetric storage devices, without energy differential only the Shannon entropy/information changes,
\[W\geq -k_BT[ H(M') - H(M)]\]
Thus resetting memory back from zero-information state requires (Landauer's principle)
\[W_{erase}\geq k_BT H(M)\]
Stable Memory & Landauer's Principle
非平衡统计物理读书会 - 信息热动力学
Information Reservoirs - Maxwell's Refrigerator
Mandal, Zarzynski (2012), 10.1073/pnas.1204263109
Mandal, Quan, Zarzynski (2013), 10.1103/PhysRevLett.111.030602
非平衡统计物理读书会 - 信息热动力学
Non-equilibrium free energy without interaction is (eq.11)
\[\mathcal F(XM)=\mathcal F(X) + \mathcal F(M) + k_BT\,I(X;M)\]
Measuring non interacting system requires (eq.13)
\[W_\text{measurement}\geq\Delta\mathcal F_{total} =\Delta\mathcal F_{memory} + kT\,I(X;M)\]
RHS cancels out for symmetric memory set-up discussed before! => Error-free measurement can be done with \(W\geq0\)!
Cost of Measurement
非平衡统计物理读书会 - 信息热动力学
Feedback that depends on the memory bounds free energy change to be
\[\Delta \mathcal F_\text{feedback} = \Delta \mathcal F - k_BT\,I(X;M)\]
Thus the extracted work W over the full cycle satisfies
\[W_\text{measurement} + W\geq\Delta\mathcal F_\text{memory} + \Delta \mathcal F\]
Since resetting reverses the memory effort \(\Delta\mathcal F_\text{reset} = -\Delta\mathcal F_\text{memory}\),
\[W_\text{measurement} + W_\text{reset}\geq k_BT\,I(X;M)\]
Cost of Measurement
非平衡统计物理读书会 - 信息热动力学
with Continuous Information Flow
If information exchange between subsystems
continuously, we can analyze entropy
production rate instead.
Consider subsystems \(X,Y\) as in figure (eq.15)
\[\dot S_{XY} = \dot S_X + \dot S_Y\]
\[\dot S_X = \mathrm d_t S_X + \dot S_X^\text{reservoir} - k_B \dot I_X \geq 0\]
\[\dot S_Y = \mathrm d_t S_Y + \dot S_Y^\text{reservoir} - k_B \dot I_Y \geq 0\]
Here \(\dot S^\text{reservoir}\) indicating rates of entropy change in reservoir: e.g., \(\dot S^\text{res} = -\dot Q / T\) for isothermal process.
Horowitz, Esposito (2014), 10.1103/PhysRevX.4.031015
非平衡统计物理读书会 - 信息热动力学
with Continuous Information Flow
Importantly, \(\dot I\) represent the time variation of the
mutual information
\[\dot I_X = \sum_{x\geq x'; y} p({x\rightarrow x'})\log\frac{p(y|x)}{p(y|x')}\]
\[\dot I_Y = \sum_{x;y\geq y'} p({y\rightarrow y'})\log\frac{p(x|y)}{p(x|y')}\]
Here \(p(a\rightarrow b)\) represents the transition current (see reference).
\(\dot I_X>0\) implies X is measuring Y; \(\dot I_X<0\) implies X is erasing information about Y, i.e. resetting.
Horowitz, Esposito (2014), 10.1103/PhysRevX.4.031015
非平衡统计物理读书会 - 信息热动力学
Example Flow Systems
Horowitz, Esposito (2014), 10.1103/PhysRevX.4.031015
非平衡统计物理读书会 - 信息热动力学
(a) Quantum-dot Maxwell demon
(b) Transmembrane sensor with
adaptive rate/stability
(c) Transmembrane sensor that
promotes protein with switch
(d) Brownian particle in switchable
energy landscapes
Outlook and Perspectives
非平衡统计物理读书会 - 信息热动力学
Generalizations to Consider
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Key quantity relevant for repeated measurement & feedback:
- Transfer entropy: captures new information per measurement
- Thermodynamic costs as a function of memory architecture:
- Single-memory multiple-rewrite vs. multiple-memory interactions.
- Fluctuation theorems extended to systems without detailed balance:
- autonomous Maxwell's demon and ratchets.
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optimal information motors can give rise to different entropy production compared to chemical motors[25].
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Quantum Szilárd engine models and quantum fluctuation theorems.
非平衡统计物理读书会 - 信息热动力学
Applications for Biological Sensors
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Copolymerization (DNA/RNA replication/transcription):
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Thermodynamic accuracy linked to entropic forces.
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Kinetic proofreading:
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Trade-off between accuracy and dissipation.
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Biological implications for transcription kinetics.
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Sensing & adaptation:
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Energy required to gather environmental information.
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Minimum information needed for robust adaptation.
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Predictive information sets minimal work for forecasting environmental fluctuations.
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非平衡统计物理读书会 - 信息热动力学
Open Questions
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Fundamental debates:
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Connection between thermodynamic and psychological arrows of time [93].
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Unifying Framework:
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A comprehensive theory that works for diverse processes.
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Revisiting Maxwell’s original challenge:
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Role of information in macroscopic physics and statistical mechanics.
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Subjectivity of entropy in information-driven systems.
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非平衡统计物理读书会 - 信息热动力学
open-loop phenomenon vs close-loop control
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Inspiration related to my own work: for microscopic biological machinery (e.g. cilia/flagella molecular motor machines), can information-aware theory distinguish between open-loop 'mechanics-only' phenomenon and tightly regulated close-loop feedback mechanisms?
非平衡统计物理读书会 - 信息热动力学
信息热动力学 - 集智
By Feng Ling
