# YouTube-GDD: A challenging gun detection dataset with rich contextual
information

Yongxiang Gu, , Xingbin Liao and Xiaolin Qin This work was supported in part
by Sichuan Science and Technology Program (2019ZDZX0006, 2020YFG0010), the
West Light Foundation of Chinese Academy of Sciences, National Academy of
Science Alliance Collaborative Program (Chengdu Branch of Chinese Academy of
Sciences - Chongqing Academy of Science and Technology) and Science and
Technology Service Network Initiative (KFJ-STS-QYZD-2021-21-001).
(Corresponding author: Xiaolin Qin.)The authors are with Chengdu Institute of
Computer Applications, Chinese Academy of Sciences, China and University of
Chinese Academy of Sciences, Beijing, China (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>qinxl2001@126.com).

###### Abstract

An automatic gun detection system can detect potential gun-related violence at
an early stage that is of paramount importance for citizens security. In the
whole system, object detection algorithm is the key to perceive the
environment so that the system can detect dangerous objects such as pistols
and rifles. However, mainstream deep learning-based object detection
algorithms depend heavily on large-scale high-quality annotated samples, and
the existing gun datasets are characterized by low resolution, little
contextual information and little data volume. To promote the development of
security, this work presents a new challenging dataset called YouTube Gun
Detection Dataset (YouTube-GDD). Our dataset is collected from 343 high-
definition YouTube videos and contains 5000 well-chosen images, in which 16064
instances of gun and 9046 instances of person are annotated. Compared to other
datasets, YouTube-GDD is ”dynamic”, containing rich contextual information and
recording shape changes of the gun during shooting. To build a baseline for
gun detection, we evaluate YOLOv5 on YouTube-GDD and analyze the influence of
additional related annotated information on gun detection. YouTube-GDD and
subsequent updates will be released at https://github.com/UCAS-GYX/YouTube-
GDD.

###### Index Terms:

Dataset, YOLOv5, object detection, gun detection

##  I Introduction

With the continuous development of artificial intelligence, various AI
technologies have been applied to an endless stream, among which object
detection technology has been widely used in the fields of public security,
autonomous driving, virtual reality and so on[1]. As an important component in
the field of security, video surveillance has gradually developed from
traditional manual detection to automation and intelligence in view of the low
efficiency and accuracy of manual inspection of dangerous guns in the current
surveillance videos[2].

Compared to general object detection tasks, gun detection is more challenging
since the person holding the gun is always trying to hide it completely or
partially. With recent advances in deep learning, this task is expected to
succeed[3]. In the past decade, convolutional neural networks (CNN) and
general deep learning models have been widely used in automatic object
detection. These models do not need to design features by hand, but learn
features when training models on large-scale samples. On the whole, CNN-based
detectors achieved the long-term state-of-the-art performance, while emerging
Transformer-based detectors show impressive performance and great potentiality
of application.

Anyhow, it is generally accepted that the quality of training samples imposes
the performance upper bound of deep learning-based methods[4, 5].
Unfortunately, according to our investigation, while object detection
algorithms have advanced dramatically in the past few years, the development
of public available gun datasets has not kept pace. Specifically, images of
gun datasets are often collected through image retrieval of search engines.
These samples are characterized by low image quality and the guns are often
shown in close-up view. It means that existing data can provide limited
supervision information since they are easy to recognize. What’s more, the
distribution of existing datasets is far away from the distribution of
surveillance videos, causing the limited application prospect.

To promote the development of security, this work presents a new challenging
dataset called YouTube Gun Detection Dataset (YouTube-GDD). Apart from
annotating the instances of gun, instances of person are also annotated to
provide richer supervision information. Compared to other datasets, YouTube-
GDD is ”dynamic”, containing rich contextual information and recording shape
changes of the gun during shooting.

The remainder of this paper is organized as follows. Section 2 gives a brief
analysis of the related work. Section 3 introduces the process of creating
YouTube-GDD and shows the statistic analysis. Section 4 gives the experimental
results of YOLOv5 on YouTube-GDD and finally Section 5 conclusions.

##  II Related Work

###  II-A Modern Object Detection Systems

Look back to modern object detection systems, its frameworks can be mainly
divided into two categories: two-stage and one-stage. The former is
represented by Faster RCNN[6] and the latter by YOLO[7]. The major difference
between two frameworks in structure is that the two-stage uses the sub network
of region proposals (RPN) to assist in generating proposals, while the one-
stage generates proposals directly on the feature map. A modern detector is
usually composed of following parts: a backbone which outputs the feature map
of the whole image, a neck or named FPN[8] which fuses the feature maps of
different scales to obtain multi-scale features (optional), and a head which
is used to predict classes and bounding boxes of objects based on the
proposals. In practice, a backbone is commonly pre-trained on ImageNet and a
neck is usually composed of several bottom-up paths and several top-down
paths[9].

In both two-stage and one-stage object detection algorithms, anchors are
introduced in advance to roughly surround objects with preset anchors with
different scales, length-width ratios and positions. Recently, there emerged
some anchor-free object detection methods[10, 11]. This kind of method does
not locate the object by fine-tuning the anchor, but directly regresses the
key points of the object, training and predicting the position and size of the
bounding box. Compared with one-stage detection framework, the object
detection framework based on key points has almost no disadvantage in
detection speed, but can achieve good detection performance.

Transformer is one of the most impressive technologies of recent years. It was
originally developed in natural language processing[12] and has recently been
applied in computer vision. In object detection, Transformer is used in
feature extraction networks to predict precise bounding boxes and their
corresponding categories by establishing a wider range of dependencies with
self-attention. To the best of our knowledge, DETR[13] was the first to use
the standard encoder-decoder architecture of Transformer for the object
detection task, while Swin Transformer v2[14] built a full Transformer
backbone and achieved the state-of-the-art performance on COCO[15].

###  II-B Gun Detection Algorithms

The detection of such dangerous objects as guns has always been one of the
important research subjects in the field of security. Roberto[16] introduced
Faster RCNN for automatic pistol detection and achieved impressive results on
the proposed dataset, providing zero false positives, 100% recall.
Furthermore, Rana[17] chose MobileNetV2[18] as the feature extractor in Faster
RCNN in order to adapt to mobile gun detection applications. Abdul[19] took
the advantage of the latest models such as YOLOv5s to achieve effective
results and speed. What’s more, pre-processing technique of blurring the
background with gaussian blur was used to improve F1-score. Apart from using
the graphical appearance of the weapon, Alberto[20] leveraged the extra human
pose information to improve performance of the handgun detector. Specifically,
heatmap-like images that represent pose are integrated with grayscale images
outputted by the basic handgun detector. Review these works, current gun
detection algorithms are still mainly based on the general object detection
algorithm. The improvement approach of specific scenarios requires in-depth
analysis of samples, we hope a new challenging dataset can inspire
innovations.

###  II-C Gun Detection Dataset

There have been some works on gun datasets. Pistol detection dataset[16]
contains 3000 images of short guns with rich context in the background. The
images selected from the internet contain one or more handguns in diverse
situations including video surveillance contexts. Sohas weapon detection
dataset[21] contains 3255 images, which is formed by weapons and small objects
that are handled in a similar way. It includes six different objects such as
pistols, knifes, bills, purses, smartphones and cards. With the increasing
resolution of video surveillance, high-resolution samples are required.
Considering that gathering real data is tough and time-consuming, annotations
and images of Synthetic Gun Detection Dataset[22] are generated within a 3D
environment and animated using 3D game engines. It is the largest open
synthetic gun detection dataset, containing 25,000 images of indoor scenes and
10,000 images of outdoor scenes. Although the number of Synthetic Gun
Detection Dataset is large, there is inevitably a distribution gap between the
real world.

Based on these considerations, we turned our attention to gun-related videos
on YouTube. With the rise of video platforms in recent years, people are
willing to share their life and fun through videos, including shooting records
and gun-related popular science introductions. Thanks to the temporal nature
of the video medium, gun-related videos can record different views of the same
gun and the shape changes of gun during shooting. It means that these
annotated images can provide rich contextual information and structure
information of guns.

##  III YouTube-GDD

###  III-A Image Collection

YouTube is adopted as the only source to collect gun-related images. Firstly,
we collected about 500 videos by searching for ”gun”, ”shooting”, ”weapons
show”, etc., and prioritized 720p or the highest resolution available for
download. Then, frames are splitted at twice the current frame rate for each
video, i.e., if the current video frame rate is 30, frames 60, 120, 180 and so
on will be captured. For adjacent frames, some static and repetitive fragments
are filtered out by calculating the difference of pixel values and setting the
threshold. Finally, we manually deleted all images about shooting games. All
images are saved in JPG format and the name format of each image and
corresponding label is set as ”YouTube id_original frame rate_split frame
rate_ID”.

###  III-B Image Annotation

LabelImg is used to annotate instances of gun and person in YOLO format[7].
All guns, including handguns, rifles, machine guns, etc., are uniformly
labeled as “gun”. The annotation criterion follows the coarse-to-fine rule. At
the first stage, workers are required to label four categories i.e,
”person”,”gun”,”uncertain person” and ”uncertain gun”. When annotating,
workers are required to merely rely on image information instead of video
information. At the second stage, uncertain objects at the first stage will be
judged by experts. Finally, YouTube-GDD contains two categories, namely
”person” and ”gun”, corresponding to category ids 0 and 1, respectively.

###  III-C Dataset Statistics

On the whole, our dataset is collected from 343 high-definition YouTube videos
and contains 5000 well-chosen images, in which 16064 instances of gun and 9046
instances of person are annotated. Examples of YouTube-GDD are shown in Fig. 1

Figure 1: Samples of annotated images in the YouTube-GDD dataset. Examples for
each of the first four lines are captured from the same video while examples
of the last line are captured from different videos.

After annotating, we need to divide the entire dataset into training set,
validation set and test set. Since the general division ratio is 8:1:1, we
split the entire dataset into 10 non-overlapping folds by filename, each
containing 500 images. The following indicators are concerned in YouTube-GDD:
(1) The number of person and gun instances; (2) The number of objects with
different scales; (3) How many videos does each fold came from. Following the
standard of COCO, an object with pixels less than $32\times 32$ is defined as
a small target, an object with pixels between $32\times 32$ and $96\times 96$
is defined as a medium object, an object with pixels more than $96\times 96$
is defined as a large object. Detailed statistical data are shown in Tab. I.
It should be noted that source video files of adjacent folds may have one
video overlap.

TABLE I: Statistic Information of YouTube-GDD Split | Instance | Scale |  Videos
---|---|---|---
person |  gun |  small |  middle |  large
fold1 | 467 | 1265 | 373 | 235 | 1124 | 35
fold2 | 430 | 620 | 4 | 84 | 962 | 34
fold3 | 466 | 905 | 39 | 259 | 1073 | 31
fold4 | 427 | 751 | 5 | 124 | 1049 | 31
fold5 | 471 | 716 | 11 | 120 | 1056 | 36
fold6 | 415 | 718 | 13 | 122 | 998 | 43
fold7 | 394 | 879 | 67 | 193 | 1013 | 42
fold8 | 475 | 636 | 1 | 60 | 1050 | 34
fold9 | 460 | 589 | 3 | 57 | 989 | 33
fold10 | 518 | 953 | 37 | 281 | 1153 | 32
all | 9046 | 16064 | 1106 | 3070 | 20934 | 343

Specifically, we firstly calculate the ratio of the different scales of
bounding box in the entire dataset as the probability distribution, and then
calculate the scale distribution of the bounding box of each fold. The two
folds with the lowest JS divergence are chosen as test set and validation set,
i.e, fold7 is chosen as the test set and fold6 is chosen as the validation set
while the rest are chosen as the training set.

##  IV Experiment

###  IV-A Experiment Setting

All experiments are implemented based on the official YOLOv5 v6.0 project[23],
using YOLOv5s as the basic configuration. All models are trained on a NVIDIA
RTX3090 (24GB memory) and Pytorch version 1.9.0. COCO[15] is used to pre-train
the backbone and feature pyramid network.

Consistent with the default settings of YOLOv5 project, the input image size
is set as 640×640, initial learning rate as 0.01, momentum as 0.937,
attenuation coefficient as 0.0005 and batch size as 16 during the experiment.
Stochastic Gradient Descent (SGD) optimizer is adopted and training Epochs are
set to 300. During the training, the first three Epochs are preheated by
warmup algorithm. After predicting, Non-Maximum Suppression (NMS) algorithm is
used for post-processing.

###  IV-B Evaluation Metric

For evaluation, we adopt the standard COCO-style evaluation metrics, i.e.,
$AP$ and $AP_{\\_}{50}$. $AP$ is evaluated under different IoU thresholds,
ranging from 0.5 to 0.95 with an interval of 0.05. In addition, the number of
parameters and floating point operations are also counted.

###  IV-C Result Analysis

TABLE II: Experiment Results on YouTube-GDD Test Set

Method | w/ TL | w/ AoP | GFLOPs | Params | Gun | Person
---|---|---|---|---|---|---
$AP_{\\_}{50}$ | $AP$ | $AP_{\\_}{50}$ | $AP$
YOLOv5s |  |  | 15.80 | 7.01M | 67.7 | 41.0 | - | -
| $\checkmark$ | 15.81 | 7.02M | 67.9 | 41.3 | 90.3 | 75.0
$\checkmark$ |  | 15.80 | 7.01M | 75.0 | 52.0 | - | -
$\checkmark$ | $\checkmark$ | 15.81 | 7.02M | 77.3 | 52.1 | 92.4 | 81.2

  * •

∗TL means Transfer Learning and AoP means Annotations of Person.

As shown in Tab. II, we evaluate YOLOv5s on YouTube-GDD and report its
performance on test set. When training from scratch, YOLOv5s trained with
additional annotations of person can gain 0.2 and 0.3 points improvement on
$AP_{\\_}{50}$ and $AP$. With transfer learning, YOLOv5s trained with
additional annotations of person can gain 2.3 and 0.1 points improvement on
$AP_{\\_}{50}$ and $AP$ even though the initial weights were trained on COCO
which already contains annotations of person. Specifically, person is the
subject that uses guns in most cases. Therefore, annotations of person could
naturally provide additional information that could help identify the gun.
What’s more, this strategy needs negligible extra cost. Firstly, GFLOPs and
parameters increase a little when adding the person classification branch in
the detection head of the pure gun detection model. Secondly, there have
existed well-trained person detection models, which can provide pseudo
annotations for gun detection.

##  V Conclusion

We collect and publish a new challenging gun detection dataset called YouTube-
GDD. All images are captured from high-definition Youtube videos, leading to
the high quality of training samples. Then, training set, verification set and
test set are naturally divided according to different video sets, posing a
challenge to the robustness of existing object detection algorithms. What’s
more, we introduce annotations of person into gun detection, discovering that
additional related annotated information can promote gun detection
performance.

In the future, we will continue to expand YouTube-GDD with high-quality
annotated samples and optimize existing gun detection algorithms based on gun
detection characteristics.

## Acknowledgement

We thank Lab students, namely Jingyang Shan, Siqi Zhang, Yuncong Peng, Qianlei
Wang, Gang Luo, Xin Lan, Boyi Fu, Yangge Qian, for their suggestions about
improving the YouTube-GDD dataset.

## References

  * [1] Z. Zou, Z. Shi, and Y. Guo, “Object detection in 20 years: A survey,” Tech. Rep., 2019.
  * [2] O. Elharrouss, N. Almaadeed, and S. Al-Maadeed, “A review of video surveillance systems,” _Journal of Visual Communication and Image Representation_ , vol. 77, p. 103116, 2021.
  * [3] S. Shah, M. Al-Khasawneh, and M. Uddin, “Review of weapon detection techniques within the scope of street-crimes,” in _2021 2nd International Conference on Smart Computing and Electronic Enterprise (ICSCEE)_. IEEE, 6 2021, pp. 26–37.
  * [4] J. Brownlee, _Better deep learning: train faster, reduce overfitting, and make better predictions_ , 2018.
  * [5] C. Shorten and T. Khoshgoftaar, “A survey on image data augmentation for deep learning,” _Journal of big data_ , vol. 6, no. 1, pp. 1–48, 2019.
  * [6] S. Ren, K. He, R. Girshick, and J. Sun, “Faster r-cnn: Towards real-time object detection with region proposal networks,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , vol. 39, no. 6, pp. 1137–1149, 2016\.
  * [7] J. Redmon, S. Divvala, and R. Girshick, “You only look once: Unified, real-time object detection,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2016, pp. 779–788.
  * [8] T. Lin, P. Dollár, R. Girshick, K. He, B. Hariharan, and S. Belongie, “Feature pyramid networks for object detection,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2017, pp. 2117–2125.
  * [9] Y. Gu, X. Qin, Y. Peng, and L. Li, “Content-augmented feature pyramid network with light linear spatial transformers for object detection,” Tech. Rep., 2021\.
  * [10] K. Duan, S. Bai, L. Xie, H. Qi, Q. Huang, and Q. Tian, “Centernet: Keypoint triplets for object detection,” in _Proceedings of the IEEE/CVF international conference on computer vision_ , 2019, pp. 6569–6578.
  * [11] H. Law and J. Deng, “Cornernet: Detecting objects as paired keypoints,” in _Proceedings of the European conference on computer vision (ECCV)_ , 2018, pp. 734–750.
  * [12] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. Gomez, . Polosukhin, and I, “Attention is all you need,” in _Advances in neural information processing systems_ , 2017, p. 30.
  * [13] N. Carion, F. Massa, G. Synnaeve, N. Usunier, A. Kirillov, and S. Zagoruyko, “End-to-end object detection with transformers,” in _European conference on computer vision_. Cham: Springer, 8 2020, pp. 213–229.
  * [14] Z. Liu, H. Hu, and Y. Lin, “Swin transformer v2: Scaling up capacity and resolution,” Tech. Rep., 2021.
  * [15] T. Lin, M. Maire, S. Belongie, J. Hays, P. Perona, D. Ramanan, . Zitnick, and C, “Microsoft coco: Common objects in context,” in _European conference on computer vision_. Cham: Springer, 9 2014, pp. 740–755.
  * [16] R. Olmos, S. Tabik, and F. Herrera, “Automatic handgun detection alarm in videos using deep learning,” _Neurocomputing_ , vol. 275, pp. 66–72, 2018\.
  * [17] R. Alaqil, J. Alsuhaibani, B. Alhumaidi, R. Alnasser, R. Alotaibi, and H. Benhidour, “Automatic gun detection from images using faster r-cnn,” in _2020 First International Conference of Smart Systems and Emerging Technologies (SMARTTECH)_. IEEE, 11 2020, pp. 149–154.
  * [18] M. Sandler, A. Howard, M. Zhu, A. Zhmoginov, and L. Chen, “Mobilenetv2: Inverted residuals and linear bottlenecks,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2018, pp. 4510–4520.
  * [19] A. M. Q. A. A. O. A. A. M. M. A. M. H. Abdul Hanan Ashraf, Muhammad Imran, “Weapons detection for security and video surveillance using cnn and yolo-v5s,” _Computers, Materials & Continua_, vol. 70, no. 2, pp. 2761–2775, 2022. [Online]. Available: http://www.techscience.com/cmc/v70n2/44624
  * [20] A. Velasco-Mata, J. Ruiz-Santaquiteria, N. Vallez, and O. Deniz, “Using human pose information for handgun detection,” _Neural Computing and Applications_ , vol. 33, no. 24, pp. 17 273–17 286, 2021.
  * [21] Pérez-Hernández and Francisco, “Object detection binary classifiers methodology based on deep learning to identify small objects handled similarly: Application in video surveillance,” _Knowledge-Based Systems_ , vol. 194, p. 105590, 2020.
  * [22] K. Giddens. World’s first synthetic gun detection dataset from edgecase.ai. [Online]. Available: https://www.edgecase.ai/articles/worlds-first-synthetic-gun-detection-dataset-from-edgecase-ai
  * [23] J. B. e. a. GLENN J, ALEX S. ultralytics/yolov5. [Online]. Available: https://github.com/ultralytics/yolov5/releases/tag/v6.0


# The Sun within:
active processes from two-temperature models

Faezeh Khodabandehlou and Christian Maes
Department of Physics and Astronomy, KU Leuven, Belgium

###### Abstract

We propose an embedding of standard active particle models in terms of two-
temperature processes. One temperature refers to an ambient thermal bath, and
the other temperature effectively describes “hot spots,” i.e., systems with
few degrees of freedom showing important population homogenization or even
inversion of energy levels as a result of activation. As a result, the
effective Carnot efficiency would get much higher than for our standard
macroscopic thermal engines, making connection with the recent conundrum of
hot mitochondria. Moreover, that setup allows to quantitatively specify the
resulting nonequilibrium driving, useful in particular for bringing the notion
of heat into play, and making easy contact with thermodynamic features.
Finally, we observe that the shape transition in the steady low-temperature
behavior of run-and-tumble particles (with the interesting emergence of edge
states at high persistence) is stable and occurs for all temperature
differences, including close-to-equilibrium.

McLennan distribution; active matter

##  I Introduction

Heat engines operate by extracting heat from one thermal bath at a higher
temperature and delivering heat to a similar reservoir at a lower temperature,
fermi ; Jaynes1988 . Biological systems (like animal muscles, molecular motors
or simple bacteria) often operate and are able to deliver useful work at a
body temperature that is equal to or lower than the ambient temperature. In
such a situation, when blindly applying the Carnot formula, we would predict
zero, or even negative, efficiency. But obviously, those biological systems
are not Carnot engines; they are active and far from the reversible limit.
Nevertheless, the point of departure of the present paper is that delivering
work for a two-temperature engine does not require both reservoirs to be in
equilibrium, nor do we need a macroscopic or quasistatic setting. One bath can
simply consist of many independent small nonequilibrium systems or active
agents. Its corresponding temperature is then indeed only (but in some
specific sense) effective. The involved molecules are activated and produced
by chemistry, to provide the necessary fuel. Yet, the activation is not first
entirely degraded into heat, after which a Carnot engine is started. That is
why biological processes need to be fast and to have some persistence, exactly
to avoid that thermalization.

The fact that energy transfer and the related possibility to deliver work need
not be driven by differences in real temperature between thermal (equilibrium)
baths has been noted before, e.g. in conference ; edgarreview ; edgarbrowinan
; ion ; ion2 ; it suffices that the “effective” temperatures (e.g. as
calculated from the relative occupations in a multilevel system) between two
regions are different. The main point is that the chemical reaction brings a
product (molecule or subsystem) in an excited state, with a statistical
occupation of energies showing a homogenized or even inverted population with
respect to the thermal distribution at the ambient temperature. That
activation is often concentrated in only few degrees of freedom (a vibration
mode or a couple of two-level systems), effectively mimicking a high-
temperature spot.
As an example, the hydrolysis of an adenosine triphosphate (ATP) molecule
means to split high-energy phosphoanhydride bonds with a release in chemical
energy of about say 0.4 eV/molecule. Spreading that over two degrees of
freedom gives an effective temperature $T_{\text{eff}}\simeq 0.4\,$ eV/kB
which is like 20 times room temperature (where the average thermal energy per
degree of freedom equals $k_{B}T/2=0.01$ eV); see Jaynes1988  for the more
detailed calculation. In other words, instead of zero or negative, a naive
application of the Carnot formula would yield corresponding efficiencies
reaching much higher values than we are used to for our macroscopic engines.

We thus imagine a collection of such nonequilibrium “hot spots” in contact
with a heat bath at ambient (or body) temperature, creating the possibility to
deliver work, Muller1995 ; Fahimi . We have little to say here about the
specific nature of those “hot spots” in their biological environment. It is
not unnatural for Eukaryotes to associate the spots with mitochondria, but for
bacteria, there will be other centers of activity. Surely, thermal baths are
present as well, e.g. in the form of an internal watery medium in which the
agents dissipate heat and/or outside at ambient temperature. Yet, in contrast
with the usual setup with heat engines, we have various small systems (few
degrees of freedom), very much out of equilibrium, coupled to a (more
traditional) thermal bath such as a bloodstream, a water bag or an external
environment. That setup is able to perform work, even when the small system
operates in noisy surroundings where the passive elements appear to be at
temperatures equal to or lower than the ambient temperature.
To continue, however, one needs a clear thermal interpretation of these
models, which brings us to the second main point of the paper.

To insert the study of Life into the realm of statistical physics requires
understanding it as a nonequilibrium process. It demands a quantitative
description of the departure from equilibrium. Life processes typically
possess a continuously varying amplitude where the biochemical fuel or driving
can at least in principle be varied far enough to touch death, i.e.,
equilibrium and detailed balance. The latter may give a useful theoretical
reference or point of contrast for building models mimicking life processes,
e.g. , to invoke local detailed balance leading to relevant thermal
interpretations. Indeed, a problem with current models of active systems is
that there is no clear reference which can be called the equilibrium point
from which perturbation theory can start. Usually, one considers the passive
limit (such as the asymptotics of large flipping rate for run-and-tumble
particles) as the equilibrium limit. While that is formally possible, it is
physically odd. After all, the flip rate is a time-symmetric parameter and
does not directly represent any fueling or driving. The present paper launches
two-temperature models through which important representatives of active
processes get a more standard place in nonequilibrium physics.

The next Section II directly deals with models of active particles as two-
temperature processes. Section II.1 discusses the case of run-and-tumble
particles, followed by Section II.2 on active Brownian dynamics. For both, we
propose a two-temperature interpretation. We observe transitions in the
stationary behavior as a function of the system parameters such as the
effective temperature, from which robustness of the so-called shape
transition, sol ; urtrap , can be derived, as our modeling allows to extend
the phenomenon in the two-temperature setting. Furthermore, local detailed
balance is restored and, from there, a thermal characterization becomes
possible. In particular we are able to construct the (close-to-equilibrium)
McLennan distribution, as recalled in Appendix A.
For modeling a “hot spot,” Section III.1 treats an activated molecule, in
particular in the form of a molecular switch where the order of the energies
in a multilevel system gets inverted at random times, and as a flashing
vibrational mode in Section III.2. There we show how the activation indeed
raises the effective temperature, in the sense of increased uniformization of
energy occupation statistics. It also provides theoretical modeling relevant
in the discussion concerning hot mitochondria, hotspot1 ; hotspot2 ; hotspot3
; hotspot4 ; hotnick ; mit .
The paper has many natural continuations, for example concerning the details
of the extractability of work in living systems. See e.g. anand2023  for the
method of bacterial rectification, and also muscle efficiency is discussed in
many places; see e.g. Barclay . For the biological physics of population
inversion as visible from calorimetry on active systems, we refer to
activePritha ; sim . Bringing these subjects together and unified under the
governing body of nonequilibrium statistical mechanics is a theoretical
contribution to which the present paper wishes to add.

##  II Active particles

Here comes the embedding of standard models of self-locomotion in a two-
temperature setting, with one high effective temperature governing the
dynamics of an internal variable (hot spot). Note that only in Section III we
add a description of “agitated” systems to represent such “hot spots,” and we
discuss the for our purpose relevant meaning of effective temperature.
The particles of the present section are subject to overdamped dynamics,
possibly under the influence of thermal noise and a confining potential, while
the “hot spot” adds another source of (colored) noise. We do not enter into
the details of the mechanisms through which that effective heat engine
delivers work and produces motion. The few models we discuss here can be
associated to flagellar locomotion. Yet, locomotion in living material
obviously does not always involve flagellar motion. There is also crawling,
swimming, and gliding of bacteria, responding differently to the environment
and its gradients and showing a great variety of mechanisms for employing low-
entropy sources, moti . Nevertheless, we believe the theoretical foundations
of our proposal are more universal than apparent in the following (bacterial)
models. In particular, the biological interpretation of the “hot spots” is
ignored here, but for Eukaryotes it is certainly tempting to suppose that they
are represented by “hot mitochondria”, hotspot1 ; hotspot2 ; hotspot3 ;
hotspot4 ; hotnick ; mit .

###  II.1 Run-and-tumble particles

Run-and-tumble particles (RTPs) are among the best-known toy models of active
matter. They share important features with bacterial locomotion, with E. Coli
as the main representative. We refer to the by now many studies and reviews,
including bech ; fodor ; malakar .

For simplicity we consider position $x\in{\mathbb{R}}$ subject to the
dynamics,

$\gamma\,\dot{x}_{t}=v\sigma_{t}-U^{\prime}(x)+\sqrt{2\gamma\,k_{B}T}\,\xi_{t}$
(II.1)

where soon we will put $\gamma=1,k_{B}=1$. The potential $U$ is spatially
confining around the origin. The $\xi_{t}$ is standard white noise with
ambient temperature $\beta^{-1}=k_{B}T$ giving it strength. The hot spot
dynamics is represented by $\sigma_{t}=\pm 1$, which flips sign

$\sigma\rightarrow-\sigma\;\text{ with rate
}\;\;\alpha\,\exp{-\beta_{\text{eff}}\,v\,\sigma\,x}$ (II.2)

depending on the position $x$ of the walker. For $\beta_{\text{eff}}=0$, the
$\sigma_{t}$ represents standard dichotomous noise, and then (II.1) (often
with $T=0$) describes a one-dimensional run-and-tumble motion with tumbling
rate $\alpha$. Jointly, in any event, $(x_{t},\sigma_{t})$ is a Markov process
with one continuous and one discrete variable, respectively, coupled to the
heat bath at temperature $T$ and the internal heat bath at effective inverse
temperature $\beta_{\text{eff}}$. Fig. 1 gives an impression of the situation
we wish to imagine more generally.

Figure 1: Cartoon of an active particle, for zooming in on the “hot spot” at
effective temperature $T_{\text{eff}}$, coupled to the locomotion system at
ambient temperature $T=T_{x}$, with $x$ the position of the overdamped
swimmer. The biological interpretation of the “hot spots” will certainly
differ per organism. For E. Coli as modeled via RTPs in the present section,
we have in mind motility proteins (the rotary engine) with power provided by a
proton flow, while for Eukaryotes, it is tempting to think about mitochondria.

At first sight, there is some problem with the tumbling rates (II.2), as they
depend on the location of the spatial origin. That is however the same origin
as already specified by the confining potential. Moreover, from bacterial
chemotaxis, spatial gradients in flipping rate are not uncommon. Furthermore,
for $\beta_{\text{eff}}\simeq 0$ (high internal temperature) that dependence
becomes very weak, and for the locomotion part (II.1), the force
$v\,\sigma_{t}$ is fully ignorant about the position. There is also something
to gain.
The great advantage of the parametrization in (II.1)–(II.2) is that we have a
clear reference $\beta_{\text{eff}}=\beta$ (keeping $\alpha\neq 0$) where the
condition of detailed balance holds. Fixing $\alpha>0$, we can now interpolate
between the purely random flipping ($\beta_{\text{eff}}=0$) and the
equilibrium dynamics ($\beta_{\text{eff}}=\beta$) via that temperature-driven
analog. At the same time, it brings the RTP–modeling within the realm of local
detailed balance ldb  from where thermal features are more easily interpreted.
There is a contact between (effectively) two thermal baths, one at inverse
(ambient) temperature $\beta$, and one at inverse (internal but effective)
temperature $\beta_{\text{eff}}$. The difference $\beta-\beta_{\text{eff}}$ in
inverse temperatures provides the driving, much as in a traditional heat
engine. In that way, we interpret the motion of RTPs as originating from a
two-temperature model, from which the usual thermodynamic interpretation of
heat flux holds; see e.g. heatcon . We work therefore under the questions of
the papers howfar ; nardini . Other effective thermal descriptions exist of
course, as e.g. in self .

####  II.1.1 Shape transition

The internal bath is given effective temperature $T_{\text{eff}}$, imagining
internal active processes carried by low-dimensional degrees of freedom (see
Section III). Here we look at the dependence on $\beta_{\text{eff}}$ for the
stationary dynamics and static fluctuations. In particular, we are interested
in observing the shape transition between an edgy and a confined regime sol ;
urtrap , as a function of the effective temperature.

The process satisfies detailed balance for $\beta_{\text{eff}}=\beta$ (equal
temperatures), with equilibrium density

$\rho_{\text{eq}}(x,\sigma)\propto e^{-\beta E(x,\sigma)},\qquad
E(x,\sigma)=-v\sigma x+U(x)$ (II.3)

at (unique) inverse temperature $\beta$ (independent of $\alpha$, as it
should), at least as long as the potential $U$ is sufficiently confining. For
example, for the harmonic potential $U(x)=x^{2}$, we have

$\rho_{\text{eq}}(x,\sigma)\propto e^{-\beta(x-v\sigma/2)^{2}}$ (II.4)

and it depends on the propulsion speed $v$ and on $\beta$ whether
$\rho_{\text{eq}}(x)=(\rho_{\text{eq}}(x,+1)+\rho_{\text{eq}}(x,-1))/2$ has
its maximum at the origin $x=0$ (confined regime) or at the edges $x=\pm v/2$
(edgy regime).

We are interested mostly in large $\beta$ (small thermal noise for
locomotion).
Suppose $0<\beta_{\text{eff}}\leq\beta$: the flipping rate
$\alpha\,\exp{-\beta_{\text{eff}}\,v\,\sigma\,x}$ is going to be small when
$\sigma\,x>0$ and $\beta_{\text{eff}}$ is large, and hence, say for
$\sigma=1$, the particle will continue to large $x>0$ without flipping
(flipping rate decreasing ever more). For example, for $\sigma\,x\sim v$, the
particle is expected to get trapped for a long time, especially at large
propulsion speed $v$, and $\alpha$ needs to be truly large to avoid that. For
example, it remains true at $T=0$, from (II.1), that $\dot{x}_{t}=0$ when
$x=\sigma\,v/\kappa$ for the harmonic potential $U(x)=\kappa x^{2}/2$. Without
confinement ($U=0$) and with little thermal noise ($\beta=10$), the
trajectories look as plotted in Fig. 2(a). That is giving rise, as for
$\beta_{\text{eff}}=0$, to edge states; compare Fig. 2(a) with Fig. 2(b) which
represents trajectories in the standard RTP-process for which
$T=0,\beta_{\text{eff}}=0$.

(a) For $n=6$ at $\beta_{\text{eff}}=0.5\beta$

(b) For $n=6$ at $\beta_{\text{eff}}=0$

(c) For $n=2$ at $\beta_{\text{eff}}=-0.1\beta$

Figure 2: Trajectories of $n$ particles under free low-temperature locomotion
(II.1)–(II.2) with $U=0$, $v=1,\beta=10,\alpha=0.5$, for different
$\beta_{\text{eff}}$, from low(a) to high(b)–(c) internal temperatures.

Suppose next that $\beta_{\text{eff}}<0$. The hot spot has reached a negative
temperature (indicating population inversion for a multilevel system). The
flipping rate $\alpha\,\exp{-\beta_{\text{eff}}\,v\,\sigma\,x}$ is going to be
small when $\sigma\,x<0$, and large when $\sigma\,x>0$. Hence, if initially
the particle is pushed with $\sigma=1$ to the positive $x>0$, it will flip a
lot and its motion remains bounded. Similarly, when $\sigma=-1$ and pushed to
negative $x<0$, it will flip a lot, and return to the origin. That implies
that for $\beta_{\text{eff}}<0$ the particles remain confined around the
origin even for relatively small $\alpha$, and even when $U=0$ (no confining
potential) at small $T$. The typical trajectories (for
$U=0,\beta=10,\,\beta_{\text{eff}}<0$) are shown in Fig. 2(c). For
$\beta_{\text{eff}}\simeq\beta$, we need a large flipping rate to see the
(passive) confined regime, as illustrated by comparing Fig. 3(a) and (b) with
Fig. 3(c). See Fig. 4 for the boundary of the two regimes.
For $\beta_{\text{eff}}>\beta$, we would have that the hidden bath has a lower
temperature than the environment, which would make changes in configuration
less efficient (as if the biological process must warm up the hidden degrees
of freedom, instead of using their energy).

(a) Edge state, visible in the stationary distribution at
$\alpha=2\quad\beta_{\text{eff}}=0.1$

(b) Edge state, visible in the stationary distribution at
$\alpha=30\quad\beta_{\text{eff}}=0.4$

(c) Confinement at $\alpha=10,\quad\beta_{\text{eff}}=0.1$

Figure 3: Stationary density of the two-temperature RTP-process at $v=3,T=0$
with confining potential $U=x^{2}$. For large enough $\alpha$, depending on
$\beta_{\text{eff}}$ and $v$, the two peaks at the edges merge and we enter
the confined regime with a single peak at the origin.  Figure 4: Shape
transition in the ($\beta_{\text{eff}},\alpha$)-plane for harmonic potential
$U(x)=x^{2}$ at $T=0$ and with propulsion speed $v=3$. The colored section
represents the edgy regime. For high enough $\alpha$ starts the confined
regime, with a sharply increasing transition line, fitting the transition as
observed numerically by coloring 40 points depending on the concavity
(colored) versus convexity of the simulated stationary density at the origin;
see also Fig. 3.

####  II.1.2 Heat

The parametrization in the two-temperature process (II.1)–(II.2) easily
identifies the driving and permits a standard discussion of thermal
properties. Since the system is open (to two baths) the energy
$E(x,\sigma)=-v\sigma x+U(x)$ changes and it does so in two ways:

$\frac{\textrm{d}}{\textrm{d}t}\langle
E(x_{t},\sigma_{t})\rangle=\big{\langle}(L_{x}+L_{\sigma})E\,(x_{t},\sigma_{t})\big{\rangle}$

for

$\displaystyle L_{x}E(x,\sigma)$ $\displaystyle=$
$\displaystyle(-v\sigma+U^{\prime}(x))\,[v\sigma-U^{\prime}(x)]+T\,U^{\prime\prime}(x)$
(II.5) $\displaystyle L_{\sigma}E(x,\sigma)$ $\displaystyle=$ $\displaystyle
2v\sigma\,x\,\alpha\,\exp{\frac{\beta_{\text{eff}}}{2}[E(x,\sigma)-E(x,-\sigma)]}$

Therefore, upon specifying the instantaneous variables $(x,\sigma)$,

$\displaystyle\frac{\textrm{d}}{\textrm{d}t}\langle
E(x_{t},\sigma_{t})\,|\,(x(0)=x,\sigma(0)=\sigma\rangle_{|_{t=0}}+\dot{q}_{x}(x,\sigma)+\dot{q}_{\sigma}(x,\sigma)=0$
(II.6) $\displaystyle\text{ for
}\qquad\dot{q}_{x}(x,\sigma)=[v\sigma-U^{\prime}(x)]^{2}-T\,U^{\prime\prime}(x)$
(II.7) $\displaystyle\hskip
23.61583pt\qquad\dot{q}_{\sigma}(x,\sigma)=-2v\sigma\,x\,\alpha\,\exp{-\beta_{\text{eff}}v\,\sigma\,x}$
(II.8)

where $\dot{q}_{x}(x,\sigma)$ equals the expected heat flux to the ambient
bath at inverse temperature $\beta$, and $\dot{q}_{\sigma}(x,\sigma)$ is the
expected heat flux to the internal bath at inverse temperature
$\beta_{\text{eff}}$. Of course, in the steady distribution, their sum
$\langle\dot{q}_{x}(x,\sigma)+\dot{q}_{\sigma}(x,\sigma)\rangle^{s}$ (expected
value in the stationary distribution) equals zero, and there is a constant
flow of energy between the two heat baths. From the above, as is common
practise now by using standard methods such as outlined in jmp ; heatcon ;
jchemphys , various thermal properties can be derived such as variable and
mean entropy production, heat capacities, and delivered work in heat
conduction networks. We only sketch, next, the less common example of
constructing the McLennan ensemble, McL ; JAM .

####  II.1.3 McLennan regime

Thanks to the two-temperature embedding, we are able to spell out the feverish
close-to-equilibrium regime, where $T$ gets closer to the effective
temperature $T_{\text{eff}}$.
We come back to the case (II.1) with confining potential and thermal noise. We
are interested in the stationary distribution for its first (linear,
nonequilibrium) correction to (II.3).
Imagining that $(\beta-\beta_{\text{eff}})\,v\,x$ remains small (no big
excursions), the considered backward generator for (II.1) has the form
$L=L_{\text{eq}}+\delta\,L_{1}$ where the perturbation
$\delta=1-\beta_{\text{eff}}/\beta$ is in the flipping part, giving the
dissipated power (A.1),

$\displaystyle p_{1}(x,\sigma)$ $\displaystyle=$
$\displaystyle-2v\sigma\,x\,\alpha\,\exp{-\beta\,v\,\sigma\,x}$ (II.9)
$\displaystyle=$
$\displaystyle-2v\,x\,\alpha\,\,[\sigma\,\cosh\beta\,\,v\,x-\sinh\beta\,\,v\,x]$

which can be inserted in the general formulæ (A.2)–(A.3), and from which we
obtain the McLennan distribution $\rho_{\text{ML}}=\rho^{\text{eq}}(1-\beta
W)$, with $W$ (of order $\delta$) from (A.2).

We already see however from (II.9), and as a good approximation to the
effective potential for small $\alpha$, that for $U(x)=x^{2}/2$, the potential
$V_{\text{ML}}(x)=\frac{x^{2}}{2}+\delta x\,\sum_{\sigma}v\,\sigma\,\alpha
e^{-\beta v\,\sigma\,x}$ may generate an edge state: we refer to Fig. 5, where
$V_{\text{ML}}(x)$ is plotted, with a minimum in $x=0$, and going to $-\infty$
for $x\rightarrow\infty$. The calculated dependence on $\alpha$ is as already
seen in Fig. 4; there are edge states arbitrarily close to equilibrium at very
low temperatures.

Figure 5: The effective potential generating the close-to-equilibrium
ensemble, showing the occurrence of edge states as already observed in Figs.
3–4.

###  II.2 Active Brownian motion

We move to two dimensions with position $\vec{r}$ for a point particle,
satisfying the equation of motion in Cartesian coordinates $\vec{r}=(x,y)$,

$\displaystyle\dot{\vec{r}}(t)$ $\displaystyle=$ $\displaystyle
v\,\hat{n}(t)-\mu\,\vec{\nabla}U(\vec{r}(t))+\sqrt{2D}\,\vec{\xi}(t)$
$\displaystyle\dot{\theta}$ $\displaystyle=$
$\displaystyle\chi[v\,x\,\sin\theta-v\,y\,\cos\theta]+\sqrt{2R}\,\zeta(t)$
(II.10)

where the unit vector $\hat{n}=(\cos\theta,\sin\theta)$ replaces the
dichotomous noise in (II.1). As before, $\vec{\xi}$ is standard white noise on
the two-dimensional locomotion of the particle, while $\zeta(t)$ is rotational
white noise, constantly redirecting the particle. The external potential $U$
is confining and $\mu>0$ is a mobility coefficient. The constant $\chi$ can be
taken positive or negative. Note that we still deal with a point particle and
the $\theta-$dynamics is indicating the direction of locomotion.
We basically repeat the same remarks and analysis of the previous section on
RTPs. The $(x,y)-$dependence in (II.2) is not visible in the locomotion (first
equation for $\dot{\vec{r}}$ ) and refers to a spatial frame where the origin
is determined by the confining potential $U$.
The ambient temperature $T$ and the internal effective temperature
$T_{\text{eff}}$ follow from

$k_{B}T=\frac{D}{\mu},\qquad k_{B}T_{\text{eff}}=\frac{R}{\chi\,\mu}$

For $\chi=0$ we get the standard active Brownian process, which, in the
present setup, refers to infinite effective temperature
$T_{\text{eff}}=\infty$ (as it was for standard run-and-tumble particles as
well); see e.g., ur2d .
In our setup and with a confining potential $U$, we imagine an energy function

$E(x,y;\theta)=-\frac{v}{\mu}x\,\cos\theta-\frac{v}{\mu}\,y\,\sin\theta+U(x,y)$

coupling the hot spot (with coordinate $\theta$) with the spatial coordinates
$(x,y)$. There is a priori no problem with taking $T_{\text{eff}}<0$ via
$\chi<0$ as that temperature is effective and may refer to population
inversion.
There are again two equilibrium cases. The persistence length is $\ell=v/R$,
reducing the dynamics to passive (or usual) Brownian motion in the limit
$\ell\downarrow 0$. Secondly, when $D\chi=R$ we get $T=T_{\text{eff}}$ and
detailed balance gets restored, with equilibrium density (for $U$ confining)

$\rho(x,y;\theta)\simeq
e^{-\beta\,E(x,y;\theta)},\qquad\beta^{-1}=k_{B}T=k_{B}T_{\text{eff}}$

We can proceed from this case, by varying $\chi$, and basically repeating the
same analysis as in Section II.1 for run-and-tumble particles. The first
dynamical phase is when $\chi\leq 0$, where the system is unconfined when
$U=0$ see Fig. 6, and the second phase happens for $\chi>0$ where the dynamics
is confined, even when $U=0$, see Fig. 7. Furthermore, the notion of heat and
the characterization of the close-to-equilibrium regime is clear, as in the
previous sections for the RTPs, since local detailed balance is verified for
all $\chi$.

Figure 6: Active Brownian motion for four particles with the dynamics (II.2)
at $\chi<0$, when $v=R=D=1$ and $U=0$ . Left: trajectories when the particles
start from $r(0)=0$. Right: the distance of the particle positions from the
origin.

Figure 7: Active Brownian motion for four particles for the dynamics (II.2)
with $\chi>0$, when $v=R=D=1$ and $U=0$ . Left: trajectories when the
particles start from $r(0)=0$. Right: the distance of the particle positions
to the origin.

##  III Hot spots and effective temperature

The issue of temperature and energy fluctuations in mesoscopic systems subject
to nonequilibrium processes is a topic of much interest; see e.g. older and
recent references in temflu . To associate an effective temperature, and
especially a high temperature, to specific agents or areas within a living
system obviously raises even more questions. It is to be understood in what
sense that temperature is measurable and compatible with other constraints,
e.g., avoiding high–temperature gradients and damage caused by a local high
temperature. Whether it can be measured (e.g. by fluorescence techniques) is
not clear, and in what sense it is at all compatible with the stability of the
morphology is also an open question. However, since some good time, the
presence of “hot spots” has been widely discussed in the specialized
literature hotspot1 ; hotspot2 ; hotspot3 ; hotspot4 ; hotnick ; mit , while
the jury is still out for deciding on the precise nature of what is
experimentally seen. The subject and scope of the present paper are clearly
much more theoretical, e.g. for allowing a consistent discussion of heat in
active particle models, using the coupling with a hot spot. Here, we add two
examples of such spots, the first being a Markov jump model of an agitated
molecule (making it a molecular switch), and the second a randomly agitated
vibrational mode, where we specify the effective temperature as a measure of
the width of the population homogenization with respect to the equilibrium
Boltzmann-Gibbs statistics.

###  III.1 Multilevel switch

As an example of an agitated molecule, we consider a switch. Molecular
switches msphillip ; msanimal ; msarch  are biologically involved in signaling
pathways. They are proteins that flip between two architectures causing either
an on\- or off-state for intracellular signaling. The on is an active state,
typically initiated from an external stimulus, mspath2021 . Other versions
include genetic toggle switches, which in turn may have practical applications
as a synthetic addressable cellular memory, Gardner2000 .
One type of modeling proceeds via specifying a multilevel system where, e.g.
at random times, energy levels are shifted. The change in energies may be
provoked by the (sudden) entrance of an ion, a proton, or even a photon that
modifies the molecular configuration or the harvesting of energy on even
smaller (nano)scales. Such processes happen on the level of individual
molecules.

A simple (and more extreme) model of a molecular switch has its energy ladder
inverted at random moments in time. We imagine an energy labeled by a
coordinate $\eta$, which is mirrored by flipping a variable $\sigma$
indicating the on\- or off-state; see Fig. 8(a). More precisely, we consider
energies

$E(\eta,\sigma)=(n-1+(n-2\eta+1)\sigma)\,\frac{\nu}{2}$ (III.1)

separated by energy unit $\nu>0$. For each $\sigma=\pm 1$, there are energy
levels $\eta=1,\ldots,n$ for a fixed integer $n\geq 2$. Typically, $n$ can be
considered small. The symmetry $E(\eta,-\sigma)=E(n-\eta+1,\sigma)$ implies
that by flipping $\sigma$ the energy ladder is simply reversed. That switch
will be responsible for a nonequilibrium occupation statistics. For $n=2$ we
are dealing with a switch for a two-level system (or classical qubit) where at
random times the ground state and the excited state get exchanged, jchemphys ;
sim .

Transitions between energy levels are described by a Master equation, for
which we introduce transition rates $k_{\sigma}(\eta,\eta\pm 1)$ (for moving
up or down in energy at given $\sigma$), and there is a constant rate
$k_{\eta}(\sigma)=\alpha$ for switching $\sigma\rightarrow-\sigma$ at a given
$\eta$. We refer to Fig.8. (b) for a graphical representation for $n=3$. Of
course, at $\eta=1$ and $\eta=n$, the system can only move up or down the
ladder (or switch).

Figure 8: Modeling a three-level switch ($n=3$). Left: switching the energy
landscape. Right: graph of transitions with an indication of energy.

In particular, we take the parameterization,

$\displaystyle k_{-}(\eta,\eta+1)$
$\displaystyle=k_{+}(\eta+1,\eta)=\frac{1}{\tau}e^{-\frac{\beta\nu}{2}}$
(III.2) $\displaystyle k_{+}(\eta,\eta+1)$
$\displaystyle=k_{-}(\eta+1,\eta)=\frac{1}{\tau}e^{\frac{\beta\nu}{2}}$
$\displaystyle k_{\eta}(\sigma)$ $\displaystyle=\alpha$ (III.3)

where $\alpha\tau>0$ indicates the switching amplitude compared to moving
up/down the ladder in Fig.8). Units of time are arbitrary here, as we only
deal with the steady condition. We have also not specified the kinetic barrier
in terms of the time-scale $\tau$ in the energy transitions (III.2); that
would introduce another dependence on $\beta$ which we skip for the present
discussion. Finally, we imagine many independent copies of that system, which
makes the analysis meaningful.

In the limit $\alpha\downarrow 0$, one disconnects the two sides of the ladder
obtaining two separate equilibrium systems. They are described by the
Boltzmann–Gibbs distribution
$\rho_{\text{eq+}}(\eta)\sim\exp-\beta(n-\eta)\nu$ and
$\rho_{\text{eq-}}(\eta)\sim\exp-\beta\,(\eta-1)\nu\big{)}$ which are of
course identical by a simple change of variable. Without agitation, the
thermal properties of the molecule are just that of an equilibrium system at
inverse temperature $\beta$.
In order to estimate the effective temperature of the system we keep the
energy-variable $\eta$ coupled to the ambient bath but we also parameterize
the switching in (III.3) via

$k_{\eta}(\sigma)=\alpha\,e^{\frac{\beta(1-\varepsilon)}{2}(E(\eta,\sigma)-E(\eta,-\sigma))}$
(III.4)

for some $\varepsilon\geq 0$. Detailed balance holds true for $\varepsilon=0$.
A the other end is the usual setup for switching, where one would take
$\varepsilon=1$ and thus have a (purely) random flipping between two energy
ladders (as in (III.3)). Then, the switch occurs at rate $\alpha$
independently of the molecular state $\eta$, and no feedback occurs.
For $\varepsilon=1$, as $\alpha\uparrow\infty$, we reach the uniform
stationary distribution

$\rho^{s}(\eta,\sigma)=\frac{1}{2n}$ (III.5)

which can be considered the infinite-temperature distribution indeed
corresponding with $\beta^{\prime}=0$. See Appendix B for the proof.
Under (III.4). the $\alpha\uparrow\infty$-limit of the stationary distribution
still depends on $\varepsilon$. As we move from $\varepsilon=0$ to
$\varepsilon=1$, always for large $\alpha$, the stationary distribution
$\rho^{s}(\eta)$ over $\eta$ becomes more and more uniform. In other words,
the nonequilibrium driving adds weight to the levels that are less occupied in
equilibrium, realizing effectively an occupation statistics corresponding to a
higher temperature (infinite upon reaching (III.5) for $\varepsilon=1$ ).

To be more specific we take the case of $n=2$, and we define the effective
temperature,

$\beta_{\text{eff}}=(k_{B}\,T_{\text{eff}})^{-1}=\frac{1}{\nu}\,\log\frac{\rho(1,-)}{\rho(2,-)}$
(III.6)

measuring the homogenization of energy occupation. In that case (of $n=2$),
there are four states and the stationary distribution is

$\displaystyle\rho(1,-)$
$\displaystyle=\rho(2,+)=\frac{e^{\beta\nu}\left(\alpha\tau+e^{\frac{\beta\varepsilon\nu}{2}}\right)}{2\left(\alpha\tau
e^{\beta\varepsilon\nu}+\alpha\tau
e^{\beta\nu}+e^{\frac{\beta\varepsilon\nu}{2}}+e^{\frac{1}{2}\beta(\varepsilon+2)\nu}\right)}$
(III.7) $\displaystyle\rho(1,+)$ $\displaystyle=\rho(2,-)=\frac{\alpha\tau
e^{\frac{\beta\varepsilon\nu}{2}}+1}{2\left(\alpha\tau
e^{\frac{\beta\varepsilon\nu}{2}}+\alpha\tau
e^{-\frac{1}{2}\beta(\varepsilon-2)\nu}+e^{\beta\nu}+1\right)}$ (III.8)

Those stationary probabilities are plotted in Fig. 9 for varying $\varepsilon$
and $\alpha$.

(a) $\beta=\nu=1,\quad\alpha\in[0,5]$

(b) $\beta=\nu=1,\quad\alpha\in[0,5]$

Figure 9: Comparing the stationary probabilities of states $(1,+)=(2,-)$ and
$(1,-)$ for the dynamics with rates (III.2)-(III.4) for $n=2$, (a) increasing
$\varepsilon<1$ brings the level probabilities closer, (b) increasing
$\varepsilon>1$ leads to population inversion, e.g. starting from
$\alpha\simeq 0.2$ for $\varepsilon=3$.

Note that increasing $\varepsilon$ uniformizes the stationary distribution for
$\varepsilon<1$, and then, for $\varepsilon>1$ and at least for large enough
$\alpha$, leads to population inversion. It implies that the effective
temperature (III.6) moves up from $T_{\text{eff}}=T$ at $\varepsilon=0$, to
$T_{\text{eff}}=\infty$ at $\varepsilon=1$, and then becomes negative for
$\varepsilon>1$ (for large enough switching rate $\alpha)$; see Fig. 10
qualifying the model as describing a hot spot for $\varepsilon\simeq 1$.

Figure 10: The effective inverse temperature (III.6) for $\beta=\nu=1$, as
function of $\alpha\in[0,5]$.

###  III.2 Agitated vibrational mode

A toy model that has been used for generating motion of particles is that of
Brownian ratchets, brownianratchet ; Astumian ; Reimann ; artibrown ; drift ;
thermalratchet ; Brownian_Parrondogame ; Brownian_gambling ; directmo . The
swimming is made possible by exploiting or creating a time-dependent
landscape. We represent that landscape again by a variable $\eta$, as denoting
the energy levels in the molecular switch of the previous section, but now it
takes continuous values and may indicate the value of a vibrational mode in a
molecule.

A simple example already appeared in activePritha ; karel  for illustrating
active calorimetry, and is obtained by taking a harmonic potential $\Phi$ on
the real line with a spring ‘constant’ $\kappa(1\pm w)$ switching values,

$\displaystyle\Phi(\eta,\sigma)=\frac{\kappa}{2}\,(1+w\,\sigma)\eta^{2},\qquad\sigma=\pm
1$

We take $\kappa>0$, but $w$ can be negative and even have $|w|>1$ to allow
higher activation. The idea is that $\kappa\,\eta^{2}/2$ is still measuring
the vibrational energy. The binary variable $\sigma$ flips at rate $\alpha>0$,
while

$\gamma\,\dot{\eta}_{t}=-\kappa\,w\,\sigma_{t}\,\eta_{t}+\sqrt{2\,\gamma\,k_{B}T}\,\xi_{t}$
(III.9)

As for an overdamped dynamics in a background at temperature $T$, we take
$\xi_{t}$ standard white noise, and we put $\gamma=1,k_{B}=1$ as before.

To get a measure of the effective temperature, we take the spreading
$\langle\eta^{2}\rangle^{s}$ in the stationary distribution (if it exists). It
indeed senses the occupation at large energies. In equilibrium ($w=0$), we
have $\langle\eta^{2}\rangle^{\text{eq}}=T/\kappa$.
It is easy enough to obtain the moment equations from (III.9),

$\displaystyle\frac{\textrm{d}}{\textrm{d}t}\langle\eta_{t}^{n}\rangle$
$\displaystyle=$
$\displaystyle-n\,\kappa\langle(1+w\sigma_{t})\eta_{t}^{n}\rangle+T\,n(n-1)\langle
x^{n-2}_{t}\rangle$ (III.10)
$\displaystyle\frac{\textrm{d}}{\textrm{d}t}\langle\sigma_{t}\,\eta_{t}^{n}\rangle$
$\displaystyle=$ $\displaystyle-\kappa
n\,\langle(\sigma_{t}+w)\,\eta_{t}^{n}\rangle+T\,n(n-1)\langle\sigma_{t}\,\eta^{n-2}_{t}\rangle-2\alpha\langle\sigma_{t}\,\eta_{t}^{n}\rangle$
(III.11)

The stationary distribution is nonGaussian in general. Yet, putting the above
equations equal to zero, we find that the stationary moments
$a_{n}={\left<\eta^{n}\right>}^{s}$ and
$b_{n}={\left<\sigma\eta^{n}\right>}^{s}$ verify recurrence formulæ

$\displaystyle a_{2n}$
$\displaystyle=(2n-1)\frac{T}{\kappa}a_{2n-2}-w\,b_{2n}$ (III.12)
$\displaystyle b_{2n}$
$\displaystyle=-(2n-1)\frac{T}{\kappa}\,\frac{z(w\,a_{2n-2}-b_{2n-2})}{1+z(1-w^{2})}$
(III.13)

in terms of the dimensionless persistence $z=\kappa/\alpha$.
When $|w|>1$, there is a critical persistence

$z_{c}=\frac{1}{w^{2}-1}$

and for $z>z_{c}$ there is no stationary distribution, karel . In particular,
the second moments equal

$\displaystyle\langle\eta^{2}\rangle^{s}=a_{2}$
$\displaystyle=\frac{1+z}{1+z(1-w^{2})}\frac{T}{\kappa}$ (III.14)
$\displaystyle\langle\sigma\,\eta^{2}\rangle^{s}=b_{2}$
$\displaystyle=-\frac{z\,w}{1+z(1-w^{2})}\frac{T}{\kappa}$ (III.15)

Of course, we see the usual equilibrium for $w=0$, with a deviation, for
$\alpha>0$, governed by $1-w^{2}$. When $|w|<1$, the effective temperature is
slightly larger than $T$. When $|w|>1$, the stationary second moment
$\langle\eta^{2}\rangle^{s}$ diverges for $z\uparrow z_{c}$, which means that
the effective temperature reaches infinite as the flip rate
$\alpha\downarrow\alpha_{c}=\kappa\,(w^{2}-1)$. Then indeed, the system
increasingly occupies very high energy states in that single vibrational mode.
It has become a “hot spot” indeed.

##  IV Conclusions

Life on our planet has been running on the temperature difference between the
Sun (hot spot of low entropy) and vast outer space. ATP and other chemical
fuels appear to recreate that nonequilibrium force inside organisms on Earth,
activating many (but tiny) open systems. They violate detailed balance, and
effectively mimic a very-high-temperature occupation statistics. When coupled
to other degrees of motion at a different ambient temperature, a two-
temperature heat engine is created (though needing to work much faster than
allowed for in the Carnot regime).
Apart from that conceptual framework, which is generally useful for
understanding efficiencies in active or bio-mimicking matter in material
science Jaynes1988 , the embedding of active particle models in two-
temperature processes allows the use of statistical thermodynamics applied to
small systems. Local detailed balance gets restored, and a clearer
prescription of heat and entropy production becomes possible, witnessed e.g.
by including a description of the close-to-equilibrium (McLennan) regime for
active particles.

The present paper has illustrated these points using simple models of
bacterial locomotion (run-and-tumble and active Brownian particles), and by
providing examples of hot spots (molecular switches and agitated vibrational
modes). The two-temperature embedding of the run-and-tumble process has in
addition shown the robustness of the shape transition between a more confined
regime and the occurrence of edge states, as monitored by the tumbling rate.

Acknowledgments: FK thanks Simon Krekels and Kasper Meerts for fruitful
discussions on the subject of RTPs and for help with the numerical work. CM
thanks Gianmaria Falasco for pointing to relevant references.

##  Appendix A McLennan distribution

To show how to be quantitative about the distance to equilibrium, we discuss
the regime close-to-equilibrium and how it retains a more universal
interpretation. Indeed, identifying the so-called McLennan ensemble governing
the close-to-equilibrium regime is not just a perturbative result. Obviously,
it is part of linear response, but it gets more interesting when there is a
physical meaning associated to that ensemble. In the same sense that the Gibbs
formalism comes with extra interpretation and can start from knowing some
basic physical quantities (where the Hamiltonian is of central importance),
the McLennan ensemble can be constructed “physically”: its central quantity is
the transient entropy flux, but it does require understanding the thermal
structure of the dynamics. To obtain the McLennan ensemble for active
processes is an application of the program described in Section II.

We remind the reader about the close-to-equilibrium ensemble when such a
regime is possible to be defined. It starts from a clear notion of an
equilibrium reference dynamics in weak contact with a thermal environment at
uniform inverse temperature $\beta$, for which equilibrium can be defined and
studied via the Boltzmann-Gibbs distribution
$\rho^{\text{eq}}(x)\sim\exp-\beta E(x)$ for energy function $E$. Next, a
driving is applied in the spirit of local detailed balance; see e.g. ldb .
That leads to identifying the transient irreversible entropy flux. More
specifically, we have a driving amplitude $\varepsilon$ and we find the
expected dissipated power $\dot{q}(x)$ when in state $x$. We need its linear
order in the perturbation, of strength $\delta$

$\dot{q}(x)=\delta\,p_{1}(x)+O(\delta^{2})$ (A.1)

to build the work function

$W(x)=\delta\,\int_{0}^{+\infty}\textrm{d}t\,\langle
p_{1}(x_{t})\rangle^{\text{eq}}_{x}$ (A.2)

where the expectation $\langle\cdot\rangle^{\text{eq}}_{x}$ is in the
equilibrium dynamics for dynamical variable $x_{t}$ (which is left unspecified
here), starting from $x_{0}=x$. The mean entropy change per $k_{B}$ in the
heat bath is given by $\beta W(x)$ when starting the equilibrium dynamics from
the system in state $x$. It is exactly that (finite) transient component of
the irreversible part of the entropy flux that corrects the Boltzmann
distribution to first order in the driving: we define the McLennan
quasipotential

$V_{\text{ML}}(x)=E(x)-W(x)$

where $E$ is the energy, and the McLennan distribution is

$\rho_{\text{ML}}(x)=\frac{1}{\mathcal{Z}}\,e^{-\beta V_{\text{ML}}(x)}$ (A.3)

For its thermal properties, the heat capacity in the McLennan ensemble was
recently studied in closeheat . That distribution is called after McLennan
because of the introduction in JAM ; see also McL .

##  Appendix B Proof of (III.5)

We prove the uniformization (III.5) by using a graphical representation of the
stationary probability law $\rho^{s}$. In general, the stationary solution of
the Master Equation for a Markov jump process can be represented by the
Kirchhoff formula

$\displaystyle\rho^{s}(\eta)=\frac{m(\eta)}{M},\quad
m(\eta)=\sum_{\mathcal{T}}m(\mathcal{T}_{\eta})\quad
m(\mathcal{T}_{\eta})=\prod_{(u,u^{\prime})\in\mathcal{T}_{\eta}}k(u,u^{\prime})$
(B.1)

where $M=\sum_{y}m(y)$, $\mathcal{T}$ is the set of all spanning trees,
$\mathcal{T}_{\eta}$ is a spanning tree rooted at $\eta$ (all edges are
directed toward $\eta$) and $m(\mathcal{T}_{\eta})$ is the weight of the
directed spanning tree $\mathcal{T}_{\eta}$; see intro ; hb .
Consider a ladder with $n$ steps and with rates given in (III.2)–(III.3),
where $\varepsilon=1$. In the limit $\alpha\uparrow\infty$, (B.1) is dominated
by the contribution of spanning trees passing over all “switching edges.” For
example, for the ladder with $n=3$ steps, the dominating spanning trees are
shown in Fig. 11.

Figure 11:  Nonoriented spanning trees of the ladder $n=3$, that dominate in
the Kirchhoff formula.

We start by showing the symmetry, when $\alpha\uparrow\infty$, for all $\eta$,

$\rho^{s}(\eta,\sigma)=\rho^{s}(\eta,-\sigma)$ (B.2)

For every tree rooted in $(\eta,\sigma)$ there exists a tree with the same
weight rooted in $(\eta,-\sigma)$. The new tree is created by changing the
direction of the edge $(\eta,-\sigma)\to(\eta,\sigma)$ into
$(\eta,\sigma)\to(\eta,-\sigma)$; see Fig. 12.

Figure 12: For every rooted tree, a new tree can be created by changing the
direction of one edge. This new tree will be rooted in the target of the
edge’s new direction. By changing the direction of the red edge, a new
spanning tree rooted at $(\eta,-\sigma)$ is created.

As a consequence, the sum over all weights of the dominating trees in
$\rho^{s}(\eta,\sigma)$ and $\rho^{s}(\eta,-\sigma)$ are equal; see(B.1).

Next we show that when $\alpha\uparrow\infty$, for all $(\eta,\sigma)$,

$\rho^{s}(\eta,\sigma)=\rho^{s}(\eta\pm 1,\sigma)$ (B.3)

Take a tree rooted at $(\eta,\sigma)$. There is a vertical edge which is
connecting $\eta\pm 1$ to $\eta$. That edge can be in $\sigma$ or $-\sigma$.
By replacing the edge to the opposite $\sigma$, and changing the direction,
the weight does not change. By changing the direction of a horizontal edge,
the weight remains the same as well. Hence, for every tree rooted at
$(\eta,\sigma)$, there exists a tree with the same weight rooted at
$(\eta+1,\sigma)$ such that the direction of one horizontal edge and of one
vertical edge are changed; see Fig. 13.

Figure 13:  By changing the direction and $\sigma$ of a vertical edge and
changing the direction of a horizontal edge, the weight of the tree is not
changing. Left: a spanning tree rooted at $(\eta,\sigma)$, Right: a new
spanning tree with the same weight rooted at $(\eta+1,\sigma)$.

Finally, from (B.2) and (B.3) we see that all states have the same stationary
probability, $\rho^{s}(\eta)=\dfrac{1}{2n}$.

## References

  * [1] E. Fermi.  Thermodynamics.  Dover publications, (1956).
  * [2] E. T. Jaynes.  The Evolution of Carnot’s Principle, pages 267–281.  Springer Netherlands, Dordrecht, (1988).
  * [3] U. Basu and C. Maes.  Nonequilibrium response and frenesy.  J. Phys. Conf. Ser., 638:012001, (2015).
  * [4] I. A. Martínez, É. Roldán, L. Dinis, and R. A. Rica.  Colloidal heat engines: a review.  Soft Matter, 13(1):22–36, (2017).
  * [5] I. A. Martínez, É. Roldán, L. Dinis D. Petrov, J. M. R. Parrondo, and R. A. Rica.  Brownian carnot engine.  Nat. Phys. , 12(1):67–70, (2015).
  * [6] R. Sarkar, I. Santra, and U. Basu.  Stationary states of activity-driven harmonic chains.  Phys. Rev. E, 107:014123, (2023).
  * [7] I. Santra and U. Basu.  Activity driven transport in harmonic chains.  SciPost Phys., 13:041, (2022).
  * [8] A. W. J. Muller.  Were the first organisms heat engines? a new model for biogenesis and the early evolution of biological energy conversion.  PBMB, 63(2):193–231, (1995).
  * [9] P. Fahimi and C. F. Matta.  The hot mitochondrion paradox: reconciling theory and experiment.  Trends Chem., 4(2):96–110, (2022).
  * [10] A. P. Solon, Y. Fily, A. Baskaran, M. E. Cates, Y. Kafri, M. Kardar, and J. Tailleur.  Pressure is not a state function for generic active fluids.  Nat. Phys, 11(8):673–678, (2015).
  * [11] U. Basu, S. N. Majumdar, A. Rosso, S. Sabhapandit, and G. Schehr.  Exact stationary state of a run-and-tumble particle with three internal states in a harmonic trap.  J. Phys. A Math. Theor., 53(9):09LT01, (2020).
  * [12] P. Fahimi and C. F. Matta.  The hot mitochondrion paradox: reconciling theory and experiment.  Trends Chem., 4(2):96–110, (2022).
  * [13] R. Dou, X. Caiand L. Ruan, J. Zhang, A. Rouzi, J. Chen, Z. Chai, and Y. Hu.  Precision nanomedicines: Targeting hot mitochondria in cancer cells.  ACS Applied Bio Materials, 5(9):4103–4117, (2022).  PMID: 36066886.
  * [14] D. Macherel, F. Haraux, H. Guillou, and O.‘Bourgeois.  The conundrum of hot mitochondria.  BBA - Bioenergetics, 1862(2):148348, (2021).
  * [15] M. A. Nasr, G. I. Dovbeshko, S. L. Bearne, N. El-Badri, and C. F. Matta.  Heat shock proteins in the “hot” mitochondrion: Identity and putative roles.  BIOESSAYS, page e1900055, (2019).
  * [16] N. Lane.  Hot mitochondria?  PLOS Biol., 16(1):1–6, (2018).
  * [17] D. Chrétien, P. Bénit, H. Ha, S. Keipert, R. El-Khoury, Y. Chang, M. Jastroch, H. T. Jacobs, P. Rustin, and M. Rak.  Mitochondria are physiologically maintained at close to $50c^{\circ}$.  PLOS Biol., 16(1):1–17, (2018).
  * [18] S. Anand, X. Ma, S. Guo, S. Martiniani, and X. Cheng.  Bacteria through obstacles: Unifying fluxes, entropy production, and extractable work in living active matter.  arXiv, 2308.08421[cond-mat.soft], (2023).
  * [19] C. J. Barclay.  Chapter 6 - efficiency of skeletal muscle.  In Jerzy A. Zoladz, editor, Muscle and Exercise Physiology, pages 111–127. Academic Press, (2019).
  * [20] P. Dolai, C. Maes, and K. Netočný.  Calorimetry for active systems.  SciPost Phys., 14:126, (2023).
  * [21] F. Khodabandehlou, S. Krekels, and I. Maes.  Exact computation of heat capacities for active particles on a graph.  J. Stat. Mech.: Theory and Experiment, 2022(12):123208, (2022).
  * [22] N. Wadhwa and H. C. Berg.  Bacterial motility: machinery and mechanisms.  Nat. Rev. Microbiol, 20:161–173, (2022).
  * [23] C. Bechinger, R. Di Leonardo, L. Hartmut, C. Reichhardt, G. Volpe, and G. Volpe.  Active particles in complex and crowded environments.  Rev. Mod. Phys., 88:045006, (2016).
  * [24] F. Étienne and M. C. Marchetti.  The statistical physics of active matter: From self-catalytic colloids to living cells.  Phys. A: Stat., 504:106–120, (2018).  Lecture Notes of the 14th International Summer School on Fundamental Problems in Statistical Physics.
  * [25] K. Malakar, V. Jemseena, A. Kundu, K. V. Kumar, S. Sabhapandit, S. N. Majumdar, S. Redner, and A. Dhar.  Steady state, relaxation and first-passage properties of a run-and-tumble particle in one-dimension.  J. Stat. Mech. Theory Exp., 2018(4):043215, (2018).
  * [26] C. Maes.  Local detailed balance.  SciPost Phys. Lect. Notes, page 32, (2021).
  * [27] C. Maes, K. Netočný, and M. Verschuere.  Heat conduction networks.  J. Stat. Phys., 111(5):1219–1244, (2003).
  * [28] É. Fodor, C. Nardini, M. E. Cates, J. Tailleur, P. Visco, , and F. van Wijland.  How far from equilibrium is active matter?  Phys. Rev. Lett., 117:038103, (2016).
  * [29] C. Nardini, E. Foder, E. Tjhung, F. van Wijland, J. Tailleur, and M. C. Cates.  Entropy production in field theories without time-reversal symmetry: Quantifying the non-equilibrium character of active matter.  Phys. Rev. X, 7:021007, (2017).
  * [30] G. Szamel.  Self-propelled particle in an external potential: Existence of an effective temperature.  Phys. Rev. E, 90:012111, (2014).
  * [31] C. Maes, F. Redig, and A. Moffaert.  On the definition of entropy production, via examples.  J. Math. Phys., 41(3):1528–1554, (2000).
  * [32] F. Khodabandehlou, C. Maes, and K. Netočný.  A Nernst heat theorem for nonequilibrium jump processes.  J. Chem. Phys., 158(20), (2023).
  * [33] C. Maes and K. Netočný.  Rigorous meaning of McLennan ensembles.  J. Math. Phys., 51(1), (2010).
  * [34] J. A. McLennan.  Statistical mechanics of the steady state.  Phys. Rev., 115:1405–1409, (1959).
  * [35] U. Basu, S. Majumdar, A. Rosso, and G. Schehr.  Active brownian motion in two dimensions.  Phys. Rev. E, 98:062121, (2018).
  * [36] Z. Fei and Y. H. Ma.  Temperature fluctuations in mesoscopic systems.  arXiv, 2309.15489 [cond-mat.Stat–Mech], (2023).
  * [37] R. Phillips.  The Molecular Switch: Signaling and Allostery.  Studies in physical biology. Princeton University Press, (2020).
  * [38] M. Hörner and W. Weber.  Molecular switches in animal cells.  FEBS Lett., 586(15):2084–2096, (2012).
  * [39] J. D. Harris, M. J. Moran, and I. Aprahamian.  New molecular switch architectures.  PNAS, 115(38):9414–9422, (2018).
  * [40] K. R. . Ghusinga, R. D. Jones, A. M. Jones, and T. C. Elston.  Molecular switch architecture determines response properties of signaling pathways.  PNAS, 118(11):e2013401118, (2021).
  * [41] T. S. Gardner, C. R. Cantor, and J. J. Collins.  Construction of a genetic toggle switch in escherichia coli.  Nature, 403:339–342, (2000).
  * [42] I. Oh, J. Song, H. R. Hyun, S. H. Lee, and J. S. kim.  “brownian ratchet for directional nanoparticle transport by repetitive stretch-relaxation of dna”.  Phys. Rev. E, 106:054117, (2022).
  * [43] R. D. Astumian.  Thermodynamics and kinetics of a brownian motor.  Science, 276(5314):917–922, (1997).
  * [44] P. Reimann.  Brownian motors: noisy transport far from equilibrium.  Phys. Rep., 361(2-4):57–265, (2002).
  * [45] P. Hänggi and Marchesoni F.  Artificial brownian motors: Controlling transport on the nanoscale.  Rev. Mod. Phys., 81:387–442, (2009).
  * [46] A. Ajdari and J. Prost.  Drift induced by a spatially periodic potential of low symmetry: Pulsed dielectrophoresis.  (1992).
  * [47] M. O. Magnasco.  Forced thermal ratchets.  Phys. Rev. Lett., 71:1477–1481, (1993).
  * [48] G. P. Harmer, D. Abbott, P. G. Taylor, and J. M. R. Parrondo.  Brownian ratchets and Parrondo’s games.  Chaos, 11(3):705–714, (2001).
  * [49] J. M. R. Parrondo and L. Dinís.  Brownian motion and gambling: from ratchets to paradoxical games.  Contemp. Phys. , 45(2):147–157, (2004).
  * [50] J. Rousselet, L. Salome, A. Ajdari, and J. Prostt.  Directional motion of brownian particles induced by a periodic asymmetric potential.  Nature, 370:446–447, (1994).
  * [51] Private communication with Karel Netočný.
  * [52] F. Khodabandehlou and C. Maes.  Close-to-equilibrium heat capacity.  arXiv, 2307.11551 [cond-mat.stat-mech], (2023).
  * [53] F. Khodabandehlou, C. Maes, and K. Netočný.  Trees and forests for nonequilibrium purposes: An introduction to graphical representations.  J. Stat. Phys, 189:41, (2022).
  * [54] C. Maes and K. Netočný.  Heat bounds and the blowtorch theorem.  Ann. H. Poincaré, 14(5):1193, (2013).