Show how to construct stochastic models which mimic the behavior of a general-purpose computer (a Turing machine). Discrete state systems obeying a Markovian master equation, which are logically reversible and have a well-defined and consistent thermodynamic interpretation

1. Thermodynamic Of
Stochastic Turing Machines
Tran Quoc Hoan
@k09hthaduonght.wordpress.com/
Paper Alert 2016-08-26, Hasegawa lab., Tokyo
The University of Tokyo
Philipp Strasberg, Javier Cerrillo, Gernot Schaller, and Tobias
Brandes, Phys. Rev. E 92, 042104 – Published 5 October 2015

2. Findings
Thermodynamic Of Stochastic Turing Machines 2
• Show how to construct stochastic models which mimic the
behavior of a general-purpose computer (a Turing machine).
• Discrete state systems obeying a Markovian master equation,
which are logically reversible and have a well-defined and
consistent thermodynamic interpretation
使うどころはよくわからん！！！

3. Turing Machines
Thermodynamic Of Stochastic Turing Machines 3
Computation T: scan
from left most blank cell to
right most blank cell
State
Symbol
Direction
0,+1,-1
Halting problem: It is impossible to design a TM TH which tells us for an
arbitrary given TM T and input s whether (T,S) will halt or not
TM

4. Stochastic Turing Machines
Thermodynamic Of Stochastic Turing Machines 4
Logically reversible computer:
principle able to unambiguously
retrace its computational path back
to the initial state.

5. Logically reversible Turing machine
Thermodynamic Of Stochastic Turing Machines 5
Based on idea of Bennett (1973)
4 tapes: input, working, history, output

6. Logically reversible Turing machine
Thermodynamic Of Stochastic Turing Machines 6
Step 1: Copy input into working tape
(state, input, working)

7. Logically reversible Turing machine
Thermodynamic Of Stochastic Turing Machines 7
Step 2: Compute
(state, working, history)

8. Logically reversible Turing machine
Thermodynamic Of Stochastic Turing Machines 8
Step 3: Copy output (in working tape) to output tape
(state, working, output)

9. Logically reversible Turing machine
Thermodynamic Of Stochastic Turing Machines 9
Step 4: Retrace computation
(state, working, history)
-> the working tape contains input
again and the history tape is returned
to blank

10. Logically reversible Turing machine
Thermodynamic Of Stochastic Turing Machines 10
Step 5: Erase working tape (state, input, working)

11. Stochastic Turing machine
Thermodynamic Of Stochastic Turing Machines 11
• Use a continuous-time Markov process to model any logically reversible TM,
simply called as stochastic TM

12. Thermodynamic of Brownian computation
Thermodynamic Of Stochastic Turing Machines 12
Approximate master
eqn. by Fokker-Planck
eqn.
Thinking about when
stochastic approximate
deterministic

13. Thermodynamic of Brownian computation
Thermodynamic Of Stochastic Turing Machines 13
Not imply that total integrated
entropy is zero

14. Thermodynamic of Brownian computation
Thermodynamic Of Stochastic Turing Machines 14
Q: How about
efficiency of
thermodynamic
computer?