# Quantropy

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Statics

#### 1.2. Dynamics

_{0}to t = t

_{1}. The action of this path is often the integral of the kinetic minus potential energy:

_{0}) and q(t

_{1}), the system will follow the path that minimizes the action subject to these constraints. This is a powerful idea in classical mechanics. However, in fact, sometimes, the system merely chooses a stationary point of the action. The Euler–Lagrange equations can be derived just from this assumption. Therefore, it is better to speak of the principle of stationary action.

_{0}at time t

_{0}and ending at a point x

_{1}at time t

_{1}, we obtain a result proportional to the amplitude for a particle to go from the first point to the second. He also gave a heuristic argument showing that as ħ → 0, this prescription reduces to the principle of stationary action.

## 2. Quantropy

Statics | Dynamics |
---|---|

statistical mechanics | quantum mechanics |

probabilities | amplitudes |

Boltzmann distribution | Feynman sum over histories |

energy | action |

temperature | Planck’s constant times i |

entropy | ??? |

free energy | ??? |

- It gives a stationary point of quantropy subject to the constraints that the amplitudes sum to 1 and the expected action takes some fixed value.
- It gives a stationary point of the free action:$$\langle A\rangle -i\mathit{\hslash}Q$$

## 3. Computing Quantropy

Statistical Mechanics | Quantum Mechanics |
---|---|

states: x ∈ X | histories: x ∈ X |

probabilities: p: X → [0, ∞) | amplitudes: a: X → ℂ |

energy: E : X → ℝ | action: A: X → ℝ |

temperature: T | Planck’s constant times i: iħ |

coolness: β = 1/T | classicality: λ = 1/iħ |

partition function: $Z={\int}_{X}{e}^{-\beta E(x)}\mathit{dx}$ | partition function: $Z={\int}_{X}{e}^{-\mathrm{\lambda}A(x)}\mathit{dx}$ |

Boltzmann distribution: p(x) = e^{−}^{βE}^{(}^{x}^{)}/Z | Feynman sum over histories: a(x) = e^{−}^{λA}^{(}^{x}^{)}/Z |

entropy: $S=-{\displaystyle {\int}_{X}p}(x)\mathrm{ln}p(x)\mathit{dx}$ | quantropy: $Q=-{\displaystyle {\int}_{X}a}(x)\mathrm{ln}a(x)\mathit{dx}$ |

expected energy: 〈E〉 = ∫_{X} p(x)E(x) dx | expected action: 〈A〉 = ∫_{X} a(x)A(x) dx |

free energy: F = 〈E〉 − TS | free action: Φ = 〈A〉 − iħQ |

$\langle E\rangle =-\frac{d}{d\beta}\mathrm{ln}Z$ | $\langle A\rangle =-\frac{d}{d\mathrm{\lambda}}\mathrm{ln}Z$ |

$F=-\frac{1}{\beta}\mathrm{ln}Z$ | $\mathrm{\Phi}=-\frac{1}{\mathrm{\lambda}}\mathrm{ln}Z$ |

$S=\mathrm{ln}Z-\beta \frac{d}{d\beta}\mathrm{ln}Z$ | $Q=\mathrm{ln}Z-\mathrm{\lambda}\frac{d}{d\mathrm{\lambda}}\mathrm{ln}Z$ |

principle of maximum entropy | principle of stationary quantropy |

principle of minimum energy (in T → 0 limit) | principle of stationary action (in ħ → 0 limit) |

## 4. The Quantropy of a Free Particle

_{i}∈ ℝ, and require that the particle keeps a constant velocity v

_{i}between the (i − 1)-st and i-th time steps:

_{0}= 0, but its final position q

_{n}is arbitrary. If we do not “nail down” the particle at some particular time in this way, our path integrals will diverge. Therefore, our space of histories is:

^{n}with coordinates q

_{1},…, q

_{n}, an obvious guess for a measure would be:

_{1}⋯dq

_{n}has units of length

^{n}. Therefore, to make the measure dimensionless, we introduce a length scale, Δx, and use the measure:

_{0}is fixed, we can express the positions q

_{1},…, q

_{n}in terms of the velocities v

_{1},…v

_{n}. Since:

_{i}are positive numbers, and we would still get the same expected action:

- Why is the expected action imaginary? The action A is real. How can its expected value be imaginary? The reason is that we are not taking its expected value with respect to a probability measure, but instead, with respect to a complex-valued measure. Recall that:$$\langle A\rangle =\frac{{\displaystyle {\int}_{X}A(x){e}^{-\mathrm{\lambda}A(x)}}\mathit{dx}}{{\displaystyle {\int}_{X}{e}^{-\mathrm{\lambda}A(x)}}\mathit{dx}}.$$The action A is real, but λ = 1/iħ is imaginary; so, it is not surprising that this “expected value” is complex-valued.
- Why does the expected action diverge as n → ∞? We have discretized time in our calculation. To take the continuum limit, we must let n → ∞, while simultaneously letting Δt → 0 in such a way that nΔt stays constant. Some quantities will converge when we take this limit, but the expected action will not: it will go to infinity. What does this mean?This phenomenon is similar to how the expected length of the path of a particle undergoing Brownian motion is infinite. In fact, the free quantum particle is just a Wick-rotated version of Brownian motion, where we replace time by imaginary time; so, the analogy is fairly close. The action we are considering now is not exactly analogous to the arc length of a path:$$\int}_{0}^{T}\left|\frac{dq}{\mathit{dt}}\right|\mathit{dt$$$${\int}_{0}^{T}{\left|\frac{dq}{\mathit{dt}}\right|}^{2}\mathit{dt}.$$However, both of these quantities diverge when we discretize Brownian motion and then take the continuum limit. The reason is that for Brownian motion, with probability 1, the path of the particle is non-differentiable, with Hausdorff dimension > 1 [6]. We cannot apply probability theory to the quantum situation, but we are seeing that the “typical” path of a quantum free particle has infinite expected action in the continuum limit.
- Why does the expected action of the free particle resemble the expected energy of an ideal gas? For a classical ideal gas with n particles in three-dimensional space, the expected energy is:$$\langle E\rangle =\frac{3}{2}nT$$$$\langle A\rangle =\frac{3}{2}ni\mathit{\hslash}.$$Why are the answers so similar?The answers are similar because of the analogy we are discussing. Just as the action of the free particle is a positive definite quadratic form on ℝ
^{n}, so is the energy of the ideal gas. Thus, computing the expected action of the free particle is just like computing the expected energy of the ideal gas, after we make these replacements:$$\begin{array}{l}E\phantom{\rule{0.3em}{0ex}}\mapsto \phantom{\rule{0.3em}{0ex}}A\\ T\phantom{\rule{0.3em}{0ex}}\mapsto \phantom{\rule{0.3em}{0ex}}i\mathit{\hslash}\end{array}$$

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Baez, J.C.; Pollard, B.S. Quantropy. *Entropy* **2015**, *17*, 772-789.
https://doi.org/10.3390/e17020772

**AMA Style**

Baez JC, Pollard BS. Quantropy. *Entropy*. 2015; 17(2):772-789.
https://doi.org/10.3390/e17020772

**Chicago/Turabian Style**

Baez, John C., and Blake S. Pollard. 2015. "Quantropy" *Entropy* 17, no. 2: 772-789.
https://doi.org/10.3390/e17020772