stochastic thermodynamics -...
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Stochastic Thermodynamics
1. Prelude & Brief History of Fluctuation theorems
2. Thermodynamics & Jarzynski/Crooks FT
3. Experiments & Applications
4. Stochastic thermodynamics & FTs
5. Detailed balance & Fluctuation-dissipation theorem
6. Information thermodynamics
7. Summary $ Outlook
Nonequilibrium statistical physics & active matter systems IOP, CAS, Beijing (Aug. 11-13, 2016)
Hyunggyu Park 박 형 규 朴 炯 奎
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Summary of Part I
Jarzynski equality
Crooks relation
: time-reverse path
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Stochastic thermodynamics
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Stochastic thermodynamics
Microscopic deterministic dynamics
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Stochastic thermodynamics
Microscopic deterministic dynamics
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Stochastic thermodynamics
Microscopic deterministic dynamics
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Stochastic thermodynamics
Microscopic deterministic dynamics
Stochastic dynamics
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Stochastic thermodynamics
Microscopic deterministic dynamics
Stochastic dynamics
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Stochastic thermodynamics
Microscopic deterministic dynamics
Stochastic dynamics
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Stochastic thermodynamics
Microscopic deterministic dynamics
Stochastic dynamics
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Stochastic thermodynamics
Microscopic deterministic dynamics
Macroscopic thermodynamics
Stochastic dynamics
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Stochastic thermodynamics
Microscopic deterministic dynamics
Macroscopic thermodynamics
Stochastic dynamics
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Stochastic thermodynamics
Microscopic deterministic dynamics
Macroscopic thermodynamics
Stochastic dynamics
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Stochastic thermodynamics
Microscopic deterministic dynamics
Macroscopic thermodynamics
Stochastic dynamics
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Stochastic thermodynamics
Microscopic deterministic dynamics
Macroscopic thermodynamics
Stochastic dynamics
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Stochastic thermodynamics
Microscopic deterministic dynamics
Macroscopic thermodynamics
Stochastic dynamics
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Stochastic thermodynamics
Microscopic deterministic dynamics
Macroscopic thermodynamics
Stochastic dynamics
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Stochastic thermodynamics
Microscopic deterministic dynamics
Macroscopic thermodynamics
Stochastic dynamics
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Stochastic thermodynamics
Microscopic deterministic dynamics
Macroscopic thermodynamics
Stochastic dynamics
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Stochastic thermodynamics
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Equilibrium
Stochastic thermodynamics
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Equilibrium
Stochastic thermodynamics
state space
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Equilibrium
Stochastic thermodynamics
state space
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Equilibrium
Stochastic thermodynamics
state space
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Equilibrium
Stochastic thermodynamics
state space
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Equilibrium
Stochastic thermodynamics
state space
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Equilibrium
Stochastic thermodynamics
state space
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Equilibrium Nonequilibrium Process
Stochastic thermodynamics
state space
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Equilibrium Nonequilibrium Process
Stochastic thermodynamics
trajectory
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Equilibrium Nonequilibrium Process
Stochastic thermodynamics
trajectory
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Equilibrium Nonequilibrium Process
Stochastic thermodynamics
trajectory
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Equilibrium Nonequilibrium Process
Stochastic thermodynamics
trajectory
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Equilibrium Nonequilibrium Process
Stochastic thermodynamics
trajectory
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Equilibrium Nonequilibrium Process
Stochastic thermodynamics
trajectory
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Equilibrium Nonequilibrium Process
Stochastic thermodynamics
trajectory
System
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Equilibrium Nonequilibrium Process
Stochastic thermodynamics
trajectory
System
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Equilibrium Nonequilibrium Process
Stochastic thermodynamics
trajectory
System
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Equilibrium Nonequilibrium Process
Stochastic thermodynamics
trajectory
System
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Equilibrium Nonequilibrium Process
Stochastic thermodynamics
trajectory
System
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Equilibrium Nonequilibrium Process
Stochastic thermodynamics
trajectory
System
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Equilibrium Nonequilibrium Process
Stochastic thermodynamics
trajectory
System
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Equilibrium Nonequilibrium Process
Stochastic thermodynamics
trajectory
System
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Equilibrium Nonequilibrium Process
Stochastic thermodynamics
trajectory
System
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Equilibrium Nonequilibrium Process
Stochastic thermodynamics
trajectory
System
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Equilibrium Nonequilibrium Process
Stochastic thermodynamics
trajectory
System
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Equilibrium Nonequilibrium Process
Stochastic thermodynamics
trajectory
System
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Langevin (stochastic) dynamics
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Langevin (stochastic) dynamics
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Langevin (stochastic) dynamics
System
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Langevin (stochastic) dynamics
state space
trajectory
System
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Langevin (stochastic) dynamics
state space
trajectory
System
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Langevin (stochastic) dynamics
state space
trajectory
System
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Langevin (stochastic) dynamics
state space
trajectory
System
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Langevin (stochastic) dynamics
state space
trajectory
System
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Langevin (stochastic) dynamics
state space
trajectory
System
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Langevin (stochastic) dynamics
state space
trajectory
System
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Langevin (stochastic) dynamics
state space
trajectory
System
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Langevin (stochastic) dynamics
state space
trajectory
System
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Langevin (stochastic) dynamics
state space
trajectory
System
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Langevin (stochastic) dynamics
state space
trajectory
System
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Langevin (stochastic) dynamics
state space
trajectory
System
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Langevin (stochastic) dynamics
state space
trajectory
System
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Langevin (stochastic) dynamics
state space
trajectory
System
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Langevin (stochastic) dynamics
state space
trajectory
System
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Langevin (stochastic) dynamics
state space
trajectory
System
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Langevin (stochastic) dynamics
state space
trajectory
System
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Stochastic process, Irreversibility & Total entropy production
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Stochastic process, Irreversibility & Total entropy productionstate space
trajectory
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Stochastic process, Irreversibility & Total entropy productionstate space
trajectory
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Stochastic process, Irreversibility & Total entropy productionstate space
trajectory
time-rev
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Stochastic process, Irreversibility & Total entropy productionstate space
trajectory
time-rev
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Stochastic process, Irreversibility & Total entropy productionstate space
trajectory
time-rev
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Stochastic process, Irreversibility & Total entropy productionstate space
trajectory
time-rev
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Stochastic process, Irreversibility & Total entropy productionstate space
trajectory
time-rev
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Stochastic process, Irreversibility & Total entropy productionstate space
trajectory
time-rev
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Stochastic process, Irreversibility & Total entropy productionstate space
trajectory
time-rev
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Total entropy production and its components
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Total entropy production and its components
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Total entropy production and its components
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Total entropy production and its components
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Total entropy production and its components
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Total entropy production and its components
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Total entropy production and its components
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Total entropy production and its components
System
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Total entropy production and its components
System
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Total entropy production and its components
System
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Total entropy production and its components
System
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Total entropy production and its components
System
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Total entropy production and its components
System
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Total entropy production and its components
System
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Total entropy production and its components
System
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Total entropy production and its components
System
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Total entropy production and its components
System
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Total entropy production and its components
System
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Total entropy production and its components
System
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Total entropy production and its components
System
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Total entropy production and its components
System
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Total entropy production and its components
System
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Total entropy production and its components
System
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Total entropy production and its components
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Total entropy production and its components
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Total entropy production and its components
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Total entropy production and its components
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Total entropy production and its components
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Total entropy production and its components
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Total entropy production and its components
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Total entropy production and its components
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Total entropy production and its components
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Total entropy production and its components
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Total entropy production and its components
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Total entropy production and its components
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Probability theory viewpoint
on Fluctuation theorems
Seifert, PRL 95, 040602 (2005) Esposito/VdBroeck, PRL 104, 090601 (2010)
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Fluctuation theorems System
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Fluctuation theorems
Integral fluctuation theorems
System
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Fluctuation theorems
![Page 116: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/116.jpg)
Fluctuation theorems
Integral fluctuation theorems
![Page 117: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/117.jpg)
Fluctuation theorems
Integral fluctuation theorems
![Page 118: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/118.jpg)
Fluctuation theorems
Integral fluctuation theorems
Thermodynamic 2nd laws
![Page 119: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/119.jpg)
Fluctuation theorems
Integral fluctuation theorems
Detailed fluctuation theorems
Thermodynamic 2nd laws
![Page 120: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/120.jpg)
Fluctuation theorems
Integral fluctuation theorems
Detailed fluctuation theorems
Thermodynamic 2nd laws
![Page 121: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/121.jpg)
Probability theory
![Page 122: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/122.jpg)
Probability theory
• Consider two normalized PDF’s :
![Page 123: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/123.jpg)
Probability theory
• Consider two normalized PDF’s : state space
trajectory
![Page 124: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/124.jpg)
Probability theory
• Consider two normalized PDF’s : state space
trajectory
![Page 125: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/125.jpg)
Probability theory
• Consider two normalized PDF’s : state space
trajectory
![Page 126: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/126.jpg)
Probability theory
• Consider two normalized PDF’s : state space
trajectory
• Define “relative entropy”
![Page 127: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/127.jpg)
Probability theory
• Consider two normalized PDF’s : state space
trajectory
• Define “relative entropy”
![Page 128: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/128.jpg)
Probability theory
• Consider two normalized PDF’s : state space
trajectory
• Define “relative entropy”
![Page 129: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/129.jpg)
Probability theory
• Consider two normalized PDF’s : state space
trajectory
• Define “relative entropy”
![Page 130: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/130.jpg)
Probability theory
• Consider two normalized PDF’s : state space
trajectory
• Define “relative entropy”
Integral fluctuation theorem
![Page 131: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/131.jpg)
Probability theory
• Consider two normalized PDF’s : state space
trajectory
• Define “relative entropy”
Integral fluctuation theorem
(exact for any finite-time trajectory)
![Page 132: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/132.jpg)
Probability theory
• Consider two normalized PDF’s : state space
trajectory
• Define “relative entropy”
Integral fluctuation theorem
(exact for any finite-time trajectory)
![Page 133: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/133.jpg)
Probability theory
• Consider two normalized PDF’s : state space
trajectory
• Define “relative entropy”
Integral fluctuation theorem
(exact for any finite-time trajectory)
![Page 134: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/134.jpg)
Probability theory
• Consider the mapping :
![Page 135: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/135.jpg)
Probability theory
• Consider the mapping :
• Require
![Page 136: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/136.jpg)
Probability theory
• Consider the mapping :
• Require reverse path
![Page 137: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/137.jpg)
Probability theory
• Consider the mapping :
• Require reverse path
![Page 138: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/138.jpg)
Probability theory
• Consider the mapping :
• Require reverse path
![Page 139: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/139.jpg)
Probability theory
• Consider the mapping :
• Require reverse path
![Page 140: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/140.jpg)
Probability theory
• Consider the mapping :
• Require
Detailed fluctuation theorem
reverse path
![Page 141: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/141.jpg)
Probability theory
• Consider the mapping :
• Require
Detailed fluctuation theorem
reverse path
(exact for any finite t)
![Page 142: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/142.jpg)
Probability theory
• Consider the mapping :
• Require
Detailed fluctuation theorem
reverse path
(exact for any finite t)
![Page 143: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/143.jpg)
Probability theory
• Consider the mapping :
• Require
Detailed fluctuation theorem
reverse path
(exact for any finite t)
![Page 144: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/144.jpg)
Probability theory
• Consider the mapping :
• Require
Detailed fluctuation theorem
reverse path
(exact for any finite t)
![Page 145: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/145.jpg)
Dynamic processes & Path probability ratio
![Page 146: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/146.jpg)
Dynamic processes & Path probability ratio
![Page 147: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/147.jpg)
Dynamic processes & Path probability ratio
![Page 148: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/148.jpg)
Dynamic processes & Path probability ratio
![Page 149: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/149.jpg)
Dynamic processes & Path probability ratio
![Page 150: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/150.jpg)
Dynamic processes & Path probability ratio
![Page 151: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/151.jpg)
Dynamic processes & Path probability ratio
![Page 152: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/152.jpg)
Dynamic processes & Path probability ratio
: time-reverse path
![Page 153: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/153.jpg)
Dynamic processes & Path probability ratio
: time-reverse path
![Page 154: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/154.jpg)
Markovian jump dynamics
![Page 155: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/155.jpg)
Markovian jump dynamics
![Page 156: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/156.jpg)
Markovian jump dynamics
![Page 157: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/157.jpg)
Markovian jump dynamics
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Markovian jump dynamics
![Page 159: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/159.jpg)
Markovian jump dynamics
![Page 160: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/160.jpg)
Markovian jump dynamics
![Page 161: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/161.jpg)
Markovian jump dynamics
![Page 162: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/162.jpg)
Markovian jump dynamics
![Page 163: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/163.jpg)
Markovian jump dynamics
![Page 164: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/164.jpg)
Markovian jump dynamics
![Page 165: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/165.jpg)
Markovian jump dynamics
![Page 166: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/166.jpg)
Markovian jump dynamics
![Page 167: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/167.jpg)
Markovian jump dynamics s R
![Page 168: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/168.jpg)
Markovian jump dynamics s R
![Page 169: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/169.jpg)
Reservoir entropy change Schnakenberg/Hinrichsen/Park
![Page 170: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/170.jpg)
Reservoir entropy change Schnakenberg/Hinrichsen/Park
![Page 171: Stochastic Thermodynamics - power.itp.ac.cnpower.itp.ac.cn/~zhouhj/Stochastic_thermodynamics-beijing-2016-II.pdf · Stochastic thermodynamics & FTs 5. Detailed balance & Fluctuation-dissipation](https://reader031.vdocuments.us/reader031/viewer/2022020316/5b4f704f7f8b9a5a6f8c6913/html5/thumbnails/171.jpg)
Reservoir entropy change Schnakenberg/Hinrichsen/Park
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Reservoir entropy change Schnakenberg/Hinrichsen/Park
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Reservoir entropy change Schnakenberg/Hinrichsen/Park
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Reservoir entropy change Schnakenberg/Hinrichsen/Park
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Reservoir entropy change Schnakenberg/Hinrichsen/Park
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Reservoir entropy change Schnakenberg/Hinrichsen/Park
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Reservoir entropy change Schnakenberg/Hinrichsen/Park
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Reservoir entropy change Schnakenberg/Hinrichsen/Park
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Reservoir entropy change Schnakenberg/Hinrichsen/Park
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Reservoir entropy change Schnakenberg/Hinrichsen/Park
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Reservoir entropy change Schnakenberg/Hinrichsen/Park
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Reservoir entropy change Schnakenberg/Hinrichsen/Park
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Langevin dynamics
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Langevin dynamics
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Langevin dynamics
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Langevin dynamics
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Discretization scheme for a path integral
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Langevin dynamics
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Langevin dynamics
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Langevin dynamics
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Langevin dynamics
: time-reverse path
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Langevin dynamics
: time-reverse path
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Langevin dynamics
: time-reverse path
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Langevin dynamics
: time-reverse path
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Langevin dynamics
: time-reverse path
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Langevin dynamics
: time-reverse path
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Langevin dynamics
: time-reverse path
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Fluctuation theorems
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Fluctuation theorems
reverse path
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Fluctuation theorems
reverse pathIrreversibility (total entropy production)
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Fluctuation theorems
reverse pathIrreversibility (total entropy production)
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Fluctuation theorems
reverse pathIrreversibility (total entropy production)
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Fluctuation theorems
reverse pathIrreversibility (total entropy production)
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Fluctuation theorems
reverse pathIrreversibility (total entropy production)
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Fluctuation theorems
reverse pathIrreversibility (total entropy production)
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Fluctuation theorems
reverse pathIrreversibility (total entropy production)
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Fluctuation theorems
reverse pathIrreversibility (total entropy production)
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Fluctuation theorems
reverse pathWork free-energy relation (dissipated work)
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Fluctuation theorems
reverse pathWork free-energy relation (dissipated work)
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Fluctuation theorems
reverse pathWork free-energy relation (dissipated work)
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Fluctuation theorems
reverse pathWork free-energy relation (dissipated work)
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Fluctuation theorems
reverse pathWork free-energy relation (dissipated work)
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Fluctuation theorems
reverse pathWork free-energy relation (dissipated work)
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Fluctuation theorems
reverse pathWork free-energy relation (dissipated work)
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Fluctuation theorems
reverse pathWork free-energy relation (dissipated work)
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Fluctuation theorems
reverse pathWork free-energy relation (dissipated work)
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Fluctuation theorems
reverse pathWork free-energy relation (dissipated work)
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Fluctuation theorems
reverse pathWork free-energy relation (dissipated work)
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Fluctuation theorems
reverse pathWork free-energy relation (dissipated work)
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Fluctuation theorems
reverse pathWork free-energy relation (dissipated work)
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Fluctuation theorems
reverse pathWork free-energy relation (dissipated work)
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Fluctuation theorems
reverse pathWork free-energy relation (dissipated work)
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Fluctuation theorems
reverse pathHouse-keeping & Excess entropy production
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Fluctuation theorems
reverse pathHouse-keeping & Excess entropy production
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Fluctuation theorems
reverse pathHouse-keeping & Excess entropy production
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Fluctuation theorems
reverse pathHouse-keeping & Excess entropy production
NEQ steady state (NESS) for fixed
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Fluctuation theorems
reverse pathHouse-keeping & Excess entropy production
NEQ steady state (NESS) for fixed
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Fluctuation theorems
reverse pathHouse-keeping & Excess entropy production
NEQ steady state (NESS) for fixed
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Fluctuation theorems
reverse pathHouse-keeping & Excess entropy production
NEQ steady state (NESS) for fixed
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Fluctuation theorems
reverse pathHouse-keeping & Excess entropy production
NEQ steady state (NESS) for fixed
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Fluctuation theorems
reverse pathHouse-keeping & Excess entropy production
NEQ steady state (NESS) for fixed
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Fluctuation theorems
reverse pathHouse-keeping & Excess entropy production
NEQ steady state (NESS) for fixed
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Fluctuation theorems
reverse pathHouse-keeping & Excess entropy production
NEQ steady state (NESS) for fixed
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Fluctuation theorems
reverse pathHouse-keeping & Excess entropy production
NEQ steady state (NESS) for fixed
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Fluctuation theorems
reverse pathHouse-keeping & Excess entropy production
NEQ steady state (NESS) for fixed