Mohammad Imran Syed
email: mohammad.syed@inria.fr

Renata Teixeira
email: renata.teixeira@inria.fr

Sara Ayoubi
email: sara.ayoubi@inria.fr

Giulio Grassi
email: giulio.grassi@inria.fr

The Challenges of Trace-Driven Wi-Fi Emulation

Keywords: H.4 [Information Systems Applications]: Miscellaneous; D.2.8 [Software Engineering]: Metrics-complexity measures, performance measures Internet measurement, Wireless link emulation, Wi-Fi record & replay

Wi-Fi link is unpredictable and it has never been easy to measure it perfectly; there is always bound to be some bias. As wireless becomes the medium of choice, it is useful to capture Wi-Fi traces in order to evaluate, tune, and adapt the different applications and protocols. Several methods have been used for the purpose of experimenting with different wireless conditions: simulation, experimentation, and tracedriven emulation. In this paper, we argue that trace-driven emulation is the most favorable approach. In the absence of a trace-driven emulation tool for Wi-Fi, we evaluate the state-of-the-art trace-driven emulation tool for Cellular networks and we identify issues for Wi-Fi: interference with concurrent traffic, interference with its own traffic if measurements are done on both uplink and downlink simultaneously, and packet loss. We provide a solid argument as to why this tool falls short of effectively capturing Wi-Fi traces. The outcome of our analysis guides us to propose a number of suggestions on how the existing tool can be tweaked to accurately capture Wi-Fi traces.

INTRODUCTION

Mobile networks are becoming increasingly more popular than wired networks due to the widespread use of mobile devices (e.g. smartphones, laptops, tablets, smartwatches, etc.). The number of smartphone users alone is expected to reach 2.87 billion by 2020 [START_REF] Department | Number of smartphone users worldwide from 2014 to 2020[END_REF].

The portability, availability, affordability, and increasing speeds of wireless connections, have made wireless the medium of choice.

The quality of wireless connectivity varies drastically from place to place depending on the coverage. There are a number of factors that affect signal quality or create interference, like poor network configuration, old equipment, fluctuating demands of users, router position, congestion, and coverage. As many of today's applications and services will be running over Wi-Fi or Cellular, it is useful to evaluate the performance of these applications in different wireless networks. For instance, application developers may wish to understand the impact of Wi-Fi packet-drop on their application, or what are the user-perceived latencies over Wi-Fi versus LTE.

There are different options for evaluating the applications and services in real network environments, namely simulation, experimentation, and emulation. Simulation is the easiest way to experiment with different wireless network conditions. Simulators are used to mimic the behavior of a certain network in a softwarebased environment. They offer different topologies, different network entities like routers, nodes, access points, etc., and tuning of real network parameters like packet loss, jitter, delay, and latency. There exist a number of wireless simulation tools [START_REF]Wikipage of NS-2 network simulator[END_REF], [START_REF]NS-3 network simulator[END_REF], [START_REF]Omnet++ network simulator[END_REF], [START_REF]Opnet network simulator[END_REF], [START_REF]NetSim Simulator by Tetcos[END_REF], [START_REF] Dinesh | Qualnet simulator[END_REF], [START_REF] Agrawal | Trace based application layer modeling in ns-3[END_REF]. Repeatability, control, configurability, and experiments of large-scale networks are the advantages of simulation. The main limitation of simulation tools, however, is that they require the user to tune different parameters e.g., level of interference, congestion, loss rate, etc. which may not reflect real wireless network conditions.

At the other end of the spectrum, there is Experimentation, where developers evaluate their applications over deployed wireless links, either over testbeds or by relying on volunteer testers. The results of such experiments capture the impact of real wireless network conditions. One disadvantage of experimentation is that it offers no repeatability. The variability of wire-1 less networks makes it hard to reproduce the results. The results of experimentation are, therefore, hard to interpret and one can not distinguish the issues with application versus wireless issues.

Finally, trace-driven emulation involves recording real wireless traces and later replaying them under emulated network conditions. The clear benefit of tracedriven emulation is its ability to capture real network conditions and the repeatability of the experiments. One can run the same network conditions several times, which eases application or system debugging, and enables comparative analysis of different applications or protocols over the same network conditions.

While there exist trace-driven emulation tools for cellular [START_REF] Winstein | Stochastic forecasts achieve high throughput and low delay over cellular networks[END_REF] and web traffic [START_REF] Netravali | Accurate record-and-replay for HTTP[END_REF], to the best of our knowledge, there exists no such tool for Wi-Fi. In this paper, we evaluate how well the state-of-the-art trace-driven emulation tool called Saturator [START_REF] Winstein | Stochastic forecasts achieve high throughput and low delay over cellular networks[END_REF], originally designed for cellular networks, works for Wi-Fi, and provide suggestions on how such a tool can be adapted to correctly record Wi-Fi traces. The rest of this paper is organized as follows: In Section 2, we highlight the existing work. Section 3 explains the challenges in using existing methods. In Section 4, we explain our measurement set-up. Section 5 presents the results of the state-ofthe-art trace-driven emulation tool [START_REF] Winstein | Stochastic forecasts achieve high throughput and low delay over cellular networks[END_REF] over Wi-Fi. In Section 6, we conclude our work and mention future work.

RELATED WORK

Simulation.

There are several network simulators. NS-2 [START_REF]Wikipage of NS-2 network simulator[END_REF] and NS-3 [START_REF]NS-3 network simulator[END_REF] are open-source network simulators to reproduce Internet systems. OMNET++ [START_REF]Omnet++ network simulator[END_REF] is a simulation platform for building simulators for wireless, wired, and queuing networks amongst others. OPNET [START_REF]Opnet network simulator[END_REF] is an open-source network simulation tool that offers various topologies and configurations. NETSIM [START_REF]NetSim Simulator by Tetcos[END_REF] is a commercial network simulator that provides simulation for layer 1 and layer 2 capabilities of Wireless Local Area Network (WLAN). QualNet [START_REF] Dinesh | Qualnet simulator[END_REF] is a commercial network simulator for scalable network technologies. It offers a Graphic User Interface (GUI) to make things easier for users as there is no coding involved. TraceReplay is an application layer simulator built in NS-3 for network traces [START_REF] Agrawal | Trace based application layer modeling in ns-3[END_REF]. Despite the fact that there are a lot of simulators available, it is always very hard to get realistic settings.

Testbeds.

Wireless Hybrid Network (WHYNET) [START_REF] Zhou | A hybrid testbed for large-scale, heterogeneous and adaptive wireless networks[END_REF] is a hybrid testbed as it allows the use of simulation, emulation, and real hardware. It allows the integration of these on both individual and combined levels. There is limited remote access to the WHYNET testbed infrastructure.

ORBIT [START_REF] Raychaudhuri | Overview of the ORBIT radio grid testbed for evaluation of next-generation wireless network protocols[END_REF] is a radio grid emulator which provides the functionality of reproducing wireless experiments with a large number of nodes. It allows the introduction of fading and controlled interference. MONROE [START_REF] Midoglu | MONROE-Nettest: A configurable tool for dissecting speed measurements in mobile broadband networks[END_REF] is an open-access hardware-based measurement platform for doing experiments on mobile broadband. The advantages of testbeds include running experiments over real wireless links and remote management. However, the testbeds come with a few drawbacks like no repeatability, no mobility (of nodes), small-level scaling, and dependency on location.

Emulation.

There are several network emulators that have been previously used to emulate network conditions for Wi-Fi and other technologies. Mobile network tracing [START_REF] Noble | Mobile network tracing[END_REF] observes traffic passively to generate traces and then uses Packet Modulator (PaM) to corrupt, delay or drop captured packets. However, mobile network tracing does not address the question of different machines sharing the same bandwidth. Trace-modulation [START_REF] Noble | Trace-based mobile network emulation[END_REF] listens to a path passively multiple times to generate traces of real network behavior. Common Open Research Emulator (CORE) [START_REF] Ahrenholz | Core: A real-time network emulator[END_REF] is a network emulator that boasts a GUI that helps in drawing topologies. While CORE emulates layer 3 and above, Extendable Mobile Ad-hoc Network Emulator (EMANE) [START_REF] Ivanic | Mobile Ad Hoc Network emulation environment[END_REF] emulates physical and data link layers (1 and 2). Mahimahi [START_REF] Netravali | Accurate record-and-replay for HTTP[END_REF] is a framework for recording and replaying HTTP traffic under different network conditions. Mahimahi uses DelayShell and LinkShell for emulating a fixed propagation delay, and fixed and variable capacity links respectively. MpShell [START_REF] Deng | or both?: Measuring multi-homed wireless Internet performance[END_REF] extends the Mahimahi framework to record Wi-Fi and LTE traces simultaneously. This work was mainly developed to evaluate the performance of MP-TCP in different network conditions and for various types of applications. A mobility emulation framework called EmuWNet [START_REF] Karimi | Measurement based mobility emulation platform for next generation wireless networks[END_REF] is proposed for signal propagation measurements in wireless networks. It allows users to replay measurement traces collected either by simulations or real-world experiments. It is based on ORBIT [START_REF] Raychaudhuri | Overview of the ORBIT radio grid testbed for evaluation of next-generation wireless network protocols[END_REF] testbed and offers various mobility scenarios for testing in a controlled environment.

In this paper, we opt for trace-driven simulation and the state-of-the-art trace-driven emulation tool, called Saturator [START_REF] Winstein | Stochastic forecasts achieve high throughput and low delay over cellular networks[END_REF]. Trace-driven emulation is a good option because 1) testing takes place on a real network and 2) traces help in repeatability; the results and testing environment can be reproduced later. We, therefore, prefer trace-driven emulation to simulation, testbeds, and experimentation.

BACKGROUND

In cellular networks, the only form of congestion at the base-station is self-induced congestion. Further, in cellular networks, the uplink and downlink communications of users take place on different time slices and they do not interfere with each other. There are rarely any standing queues created by the traffic of other users in the cell. Even in the case of individual queues, a queuing delay of 750 ms does not starve the load [START_REF] Winstein | Stochastic forecasts achieve high throughput and low delay over cellular networks[END_REF]. Whereas the medium is shared in Wi-Fi and hence, the queues at every network entity are shared by all users. Cellular networks are also more robust because there are several numbers of retransmissions to cope with packet loss, which is not the case with Wi-Fi. The throughput can only be affected by the demand and competition for allocation of the time slices in cellular networks, whereas there are other factors that affect throughput in Wi-Fi (including cross traffic). This introduces unique challenges for recording Wi-Fi traces.

We use the state-of-the-art tool Saturator [START_REF] Winstein | Stochastic forecasts achieve high throughput and low delay over cellular networks[END_REF] to demonstrate its behavior over cellular network and Wi-Fi. Saturator consists of two sender programs running at a client and a server. The client is connected via two cellphones, one cellphone is used to saturate the uplink and the downlink, while the second cellphone is used for feedback. A window of N packets is maintained by each sender program. Using the feedback packets, each sender adjusts the window size to ensure that the link is saturated without causing any self-induced packet loss. Both the client and the server record the timestamp, sequence number, and estimated round-trip time (RTT) or one-way delay for every received data or ACK packet in their respective logs. Uplink and downlink latency, throughput, and packet loss can be computed using these logs. The feedback in Saturator consists of ACK packets sent to the sender for the packets received by the receiver. The sender can then keep sending consistently to saturate the link reliably. Therefore, a separate interface is needed for feedback to ensure the timely delivery of ACK packets to the sender, and to avoid any impact of feedback delay on the link saturation. If the interface that has to be saturated, is also used for feedback, queuing might cause enough delay for ACK packets to not arrive on time. In this case, there is a possibility Network (s) Traces (s) The traces collected via Saturator are replayed using Cellsim [START_REF] Winstein | Stochastic forecasts achieve high throughput and low delay over cellular networks[END_REF]. Cellsim requires its own machine with two Ethernet interfaces. It is connected to the client directly with an Ethernet cable, whereas it is connected to the Internet with a second Ethernet interface. The client machine is not connected to the Internet, whereas the server machine is connected to the Internet via Ethernet. Cellsim delays the packets received on both its Ethernet interfaces by a considerable amount of time to emulate propagation delay before adding the packets to the queue. The traffic from the client is sent to the server over the Internet by Cellsim.

We collect the traces with Saturator for both LTE and Wi-Fi. We then use Cellsim to replay the traces captured with Saturator. We set up 3 machines for replay; one as a client, one as a server, and one as a Cellsim machine. We measure the time it takes to download a given file over a Wi-Fi network and compare the results against the time it takes to download the exact same file using traces recorded with Saturator for the same Wi-Fi network. We repeat the same measurement for LTE. To ensure that the LTE and Wi-Fi conditions do not fluctuate much between the real and trace-driven experiments, we run the real and emulated experiments back-to-back. We use a 250 MB file for the Wi-Fi experiment, and a 15 MB file for LTE, given that we have limited data for LTE.

We see in Table 1 that the file transfer completion times in both record and replay are always exactly the same for LTE. This is expected because Saturator is designed for cellular networks. Whereas, for Wi-Fi, it takes more time to complete the transfer in the replay phase. This indicates that Saturator works really well for LTE as expected but it is most probably not measuring the Wi-Fi link properly. There is likely to be some error in the Wi-Fi measurements that creates doubt about Saturator's compatibility with Wi-Fi.

EXPERIMENTAL SET-UP

In this section, we present the experimental set-up that we use for all our measurements. Figure 1 presents our measurement set-up which is as follows:

• one laptop as a client; connected to Wi-Fi for measurements and connected to an access point with an Ethernet cable for feedback. • one laptop as server; connected to the same access The AP is TP-Link Archer C7 which supports 802.11ac, and 2.4 and 5 GHz connections simultaneously. We perform our testing with 802.11ac and 2.4 GHz. We use 2 Dell laptops with identical specifications; Intel 8th generation core i7 CPU -1.9GHz (Turbo 4GHz), 16 GB RAM, and 520 GB SSD hard drive. Both machines have Ubuntu 18.04 freshly installed, they do not have anything else installed on them. We use this set-up to avoid any impact of CPU load on Saturator.

We run Saturator client and server versions on these machines. We use Linux's traffic control (tc) [START_REF] Kerrisk | Linux traffic control[END_REF] option on the client machine to vary the bandwidth and loss rate. We need to limit the bandwidth to a certain value to test Saturator's compatibility with Wi-Fi. We vary the bandwidth values every 12 seconds as follows: 15 Mbps, 40 Mbps, 10 Mbps, 30 Mbps, and 15 Mbps.

Similarly, for loss rate, we use tc and vary the percentage value for packet loss every 12 seconds as follows: 0.3%, 0.5%, 0.25%, 1%, and 0.3%.

We perform all the tests 5 times each once we have all set-up in place. The test duration is 1 minute for all the tests. We conduct tests for constant and variable bandwidth and packet loss values.

SATURATOR OVER WI-FI

In this section, we showcase the limitations of Satura- tor for Wi-Fi. Given that Wi-Fi's queuing mechanism and medium access control are different from that of Cellular, we study the impact of these two features on Saturator's ability to accurately record Wi-Fi traces. First, we evaluate Saturator's ability to saturate the Wi-Fi link without any concurrent traffic, a set-up that closely resembles Cellular links. Next, we evaluate the behavior of Saturator when we introduce concurrent UDP and TCP traffic on the Wi-Fi link. Finally, we look at the impact of saturating both the Wi-Fi uplink and downlink simultaneously.

Saturator with Concurrent Traffic

The first step is to test Saturator with Wi-Fi without any modifications and in the absence of cross-traffic; to verify if Saturator is able to saturate the link. We make use of tc to limit the bandwidth as explained in the previous section and carry out tests without any concurrent traffic.

We observe that Saturator is able to fill the pipe. Saturator reacts to the bandwidth variations very well and adapts accordingly. This is the expected behavior because the testing conditions without concurrent traffic are similar to cellular networks. Figure 2 shows the ratio of achieved throughput and capacity. The more the ratio is closer to 1, the more Saturator is able to fill the pipe. Smaller values of the ratio in Figure 2 can be deceptive because Saturator is able to consistently saturate the link. These small values of the ratio, however, are the result of changing bandwidth during the tests The main question is how would Saturator cope with concurrent traffic in Wi-Fi. We consider the following conditions while doing these tests:

• with concurrent traffic and variable bandwidth • with concurrent traffic and variable packet loss

We generate concurrent TCP and UDP traffic with iPerf [START_REF] Mah | [END_REF], and we limit the per-iPerf stream bandwidth to 5 Mbps. We see in Figure 3 that the ratio between achieved throughput and expected throughput is mostly close to 1 for Saturator; it does considerably well to fill the pipe with concurrent TCP traffic even with variable bandwidth. However, it ends up suppressing everything else, as TCP traffic gets a lot less than what is expected. As evident from Figure 3, the ratio for iPerf is way less than 1. iPerf gets around 1 Mbps for 1 stream, whereas it gets a maximum of 10 Mbps for 8 streams. It shows Saturator is not fair to TCP traffic.

The results are, however, different for concurrent UDP traffic generated by iPerf. When we use just 1 stream restricted to 5 Mbps, iPerf consistently manages to achieve the expected throughput. However, as we increase the number of streams for UDP traffic, iPerf's UDP traffic seems to saturate the pipe completely, as we observe in Figure 4. However, we see in Figure 4 that the ratio of achieved throughput and expected throughput for Saturator even exceeds 1; which suggests Saturator achieves the expected throughput. Figure 5 clears this anomaly; we see that Saturator is able to achieve the expected throughput for higher bandwidth values ). This raises a question of Saturator's compatibility with Wi-Fi. Figure 5 shows the result of one of the experiments but the results are consistent across all experiments. As discussed earlier, concurrent traffic is very likely to be present and Wi-Fi is not a stable medium as well, it casts doubt on the use of Saturator.

Another important aspect to study about Saturator is its ability to react to packet loss. We use tc to introduce packet loss as explained in Section 4.

We observe that Saturator is able to cope with the loss reasonably well as compared to the concurrent traffic. It is evident from Figures 6 and7 that it keeps sending more traffic to reach a decent throughput, as the ratio between achieved throughput and expected throughput is consistently close to 1. It is a surprising finding considering 1) it is UDP traffic and 2) Saturator does not react to packet loss itself. We examine it further to find the cause of this behavior and analyze the logs. As mentioned in Section 3, Saturator reacts to RTT only, it just keeps the check of the number of packets in flight with respect to the window size and keeps sending packets whenever there is an opportunity. As packets get lost, there are fewer packets in flight and Saturator sees it as an opportunity to send more. In this way, it ends up sending more packets than it normally does in case of no loss. The sender side has a number of packets sent more than the number of packets received at the receiver by a certain percentage in accordance with the loss.

Saturator in One Direction

As mentioned earlier, Saturator is originally designed to work with cellular network and it saturates both uplink and downlink at the same time. It does not matter much for cellular networks because both communications take place on different time and frequency slots and they do not affect each other. Whereas Wi-Fi is a shared medium and there are always chances that uplink and downlink communications happening at the same time can interfere with each other resulting in decreased performance. We, therefore, make a minor change in Saturator to make it work in only one direction. We evaluate Saturator with Wi-Fi for 2 conditions as following:

• saturate both uplink and downlink In the two-way experiments where Saturator is run in both uplink and downlink directions, it always gets more throughput on the uplink. Downlink communication is badly affected by packet loss as packet loss goes up to 80%. Figure 8 represents the packet loss for one of the five experiments, it is consistent across all five runs. Uplink packet loss for the 2-way Saturator is not clearly visible in the figure because it is exactly the same as the 1-way uplink packet loss. The client always initiates the communication, which could explain why the uplink achieves more bandwidth than the downlink. We make a slight change to make sure the client just initiates the connection with the server but it actually starts sending data with a delay of 1 second. We find out that the server takes over the bandwidth initially but as soon as the client starts sending, we see more traffic on the uplink, same as in previous cases. Multiple factors could explain this behavior; one possible explanation is that the uplink queue is larger than the downlink queue. Another possible reason is that the access point might be giving higher priority to uplink traffic than downlink traffic.

Figure 9 shows the throughputs in Mbps for one run. The results are similar for all runs. We see that the results are far better and more stable when only the uplink is saturated. The 1-way throughput is always more than the sum of uplink and downlink throughputs of the 2-way Saturator. We recommend using Saturator only in the uplink direction when used with Wi-Fi to eliminate the possible interferences and collisions between the uplink and downlink traffic and enable Saturator to fully saturate the pipe.

Packet Loss Issue

As we discussed earlier, there is a huge packet loss with Saturator running in both directions. We use the we use tcpdump [START_REF] Richardson | Tcpdump and libpcap[END_REF] to be able to capture the client's outgoing data traffic too. We use tshark [START_REF]Manpage of tshark[END_REF] to get timestamps and packet lengths from the pcap files and then use those to find the throughput. We can see in Figures 10 and11 that Saturator sends way more than what is actually received. As mentioned previously, Saturator is designed for cellular networks, it does not consider Wi-Fi network conditions and ends up sending more than what it could actually send. This results in the driver dropping the packets to cope with excessive Saturator traffic. Therefore, the packet loss we see is not because of Wi-Fi, but it is because Saturator's sending window is tuned for LTE buffer sizes, which are typically larger than Wi-Fi's.

Alternative Solutions

Given the aforementioned limitations of Saturator, in this section, we explore whether iPerf could be used instead to record Wi-Fi traces. We study the impact of the number of threads of TCP iPerf on the bandwidth it measures. We use 8 iPerf TCP streams as traffic generated by us, and a different number of streams for TCP and UDP iPerf as concurrent traffic. We consider the following scenarios:

• 8 iPerf TCP streams vs 5 iPerf TCP streams • 8 iPerf TCP streams vs 15 iPerf TCP streams • 8 iPerf TCP streams vs 5 iPerf UDP streams • 8 iPerf TCP streams vs 15 iPerf UDP streams We run tcpdump in parallel with iPerf and use the pcap files to calculate the throughput. We see the results for all these scenarios in Figures 12 and13. The bandwidth measured by iPerf depends on the number of streams used. If iPerf has more streams than concurrent traffic, then it ends up getting more bandwidth (mainly because of TCP's fair share); that would affect the real traffic in the network and create a bias in the measurements. It shows one has to be careful when choosing the number of parallel streams as it can not only create a bias in the measurements but also have a negative effect on the actual traffic; especially if multiple streams are used with UDP iPerf, it can badly degrade performance for other users.

CONCLUSION

In this paper, we evaluate how well the state-of-theart trace-driven emulation tool Saturator captures Wi-Fi traces. Since Saturator is originally designed for cellular networks, we showcase how the differences between Wi-Fi and Cellular inhibit the applicability of Saturator for Wi-Fi as-is. We highlight through experimental analysis that the measurements done by Saturator are influenced by the nature of concurrent traffic. Additionally, we illustrate how saturating Wi-Fi in just one direction eliminates interference, thereby enhancing Saturator's ability to effectively fill the pipe.

Although all the experiments we conduct are in ideal Wi-Fi conditions, for future work, we aim to repeat our experiments in different set-ups to study the effects of multipath fading. We also plan to study the impact of 802.11 frame-aggregation on trace replay. Further, we plan to explore the idea of introducing packet loss in the replay as this is a common behavior in Wi-Fi. Finally, we aim to investigate the ideal window size to eliminate the observed packet drops at the driver side.

Figure 1 :

 1 Figure 1: Experiment Set-up This figure shows the experimental set-up for our tests that includes 2 Dell laptops and a TP-Link access point

Figure 2 :

 2 Figure 2: Distribution of the fraction of capacity consumed by Saturator

Figure 3 :

 3 Figure 3: Saturator with concurrent TCP traffic and variable bandwidth

Figure 4 :

 4 Figure 4: Saturator with concurrent UDP traffic and variable bandwidthThe effect of Saturator on concurrent UDP traffic in the presence of variable bandwidth is represented in this figure

Figure 5 :

 5 Figure 5: Saturator with concurrent UDP traffic and variable bandwidth BW means bandwidth and it represents the bandwidth limit set by tc

Figure 6 :

 6 Figure 6: Saturator with concurrent TCP traffic and variable packet loss

Figure 7 :

 7 Figure 7: Saturator with concurrent UDP traffic and variable packet loss

Figure 8 :

 8 Figure 8: Saturator Packet Loss This figure represents the packet loss calculated from Saturator logs

Figure 9 :Figure 10 :

 910 Figure 9: Saturator Throughput This figure shows the comparison of 2-way Saturator throughput versus 1-way Saturator throughput

  Data Sent by Client 2way Saturator -Data Received by Server 2way Saturator -Data Sent by Server 2way Saturator -Data Received by Client

Figure 11 :

 11 Figure 11: Data sent vs data received -1-way Saturator

Figure 12 :

 12 Figure 12: Multithreaded TCP iPerf -TCP vs TCP

Figure 13 :

 13 Figure 13: Multithreaded TCP iPerf -TCP vs UDP

Table 1 :

 1 File transfer completion times in seconds

			Network (s) Traces (s)
	0.40	0.40	0.24	0.32
	0.29	0.29	0.18	0.31
	0.32	0.32	0.20	0.50
	0.37	0.37	0.24	0.30
	0.37	0.37	0.22	0.32
	(a) LTE		(b) Wi-Fi

This table shows the transfer completion time for downloading a file over LTE and Wi-Fi (Network columns) versus over-recorded and replayed LTE and Wi-Fi traces (Traces columns).


Malte Henkel

Generalised time-translation-invariance in simple ageing

These three conditions require several comments. First, the insistence on slow dynamics distinguishes systems in a disordered state with a single equilibrium state (when according to the general principles of thermodynamics, a fast relaxation within a finite characteristic time τ r towards that state should occur) from systems either at a critical point or in a two-phase coexistence region. In figure 1, the equilibrium free energy F = F(M) (in the language of magnetic systems, where M denotes the magnetisation) is shown. 1 Before the quench (left panel), the system is prepared in a disordered initial state and after a rapid relaxation, this state is found at the unique minimum of the free energy, with gaussian fluctuations around it. Immediately after the quench (right panel), the system's state has not yet evolved but is no longer at a stable mininum. At criticality (T = T c ), critical fluctuations lead to a wider form of F(M) and allow for larger and non-gaussian fluctuations which also slow down any relaxation towards equilibrium. Below criticality (T < T c ), the system now sits at an unstable maximum of F(M) and there are at least two equivalent, but distinct equilibrium states at the new minima of F(M). In both cases, a rapid relaxation towards equilibrium is impossible and the system becomes spatially inhomogeneous, mathematically expressed through the relaxation time τ r becoming formally infinite. Hence only in the situations sketched in the right panel of figure 1 ageing is at all possible. Second, the requirement of dynamical scaling asserts that the dynamics can be described in terms of a single time-dependent length scale L = L(t). To be specific, we shall restrict throughout to systems where L(t) ∼ t 1/z grows algebraically for large times, 2 which defines the dynamic exponent z. Third, the breaking of time-translation-invariance implies that ageing is an intrisically nonequilibrium phenomenon. The precise meaning of 'breaking of time-translationinvariance' will be the central topic of this work.

The relaxation process is described via the time-and space-dependent orderparameter φ = φ (t, r), conviently coarse-grained such that a continuum description becomes feasible. Unless stated otherwise, we shall assume an initial disordered state with vanishing initial order-parameter φ ini (r) = φ (0, r) = 0, where the average • is taken over the initial configurations and thermal histories. The twotime correlator C and the two-time response R are defined as [START_REF] De Dominicis | [END_REF]32]

C(t, s; r) = φ (t, r)φ (s, 0) , R(t, s; r) = δ φ (t, r) δ h(s, 0) h=0 = φ (t, r) φ (s, 0) (1)
Because of dynamical scaling, one expects, 3 for sufficiently large times t, s → ∞ and y = t/s > 1 that

C(t, s; r) = s -b F C t s ; r s 1/z , R(t, s; r) = s -1-a F R t s ; r s 1/z (2)
where a, b are ageing exponents. 4 This kind of dynamical scaling is often referred to as simple ageing. Evidence for this kind of data collapse abounds in the literature, e.g. [7,[START_REF] Cugliandolo | Slow relaxations and non-equilibrium dynamics in condensed matter[END_REF][START_REF] Henkel | Non-equilibrium phase transitions vol. 2: ageing and dynamical scaling far from equilibrium[END_REF][START_REF] Täuber | Critical dynamics: a field-theory approach to equilibrium and non-equilibrium scaling behaviour[END_REF]. If one restricts to auto-correlations and auto-responses with r = 0, one expects asymptotically for y 1

f C (y) = F C (y, 0) ∼ y -λ C /z , f R (y) = F R (y, 0) ∼ y -λ R /z (3a)
where λ C , λ R are the auto-correlation and auto-response exponents, respectively. Scale-invariance alone cannot account for such an algebraic asymptotic behaviour. Furthermore, one generally finds the exponent equality5

λ C = λ R = λ (3b)
The universality of the values of λ C , λ R and of the form of the scaling functions f C (y), f R (y) implies that they should be independent of the 'microscopic details' of any given model. In other words, we are looking for some as yet un-recognised dynamical symmetry in order to explain the assertions eqs. [START_REF] Arceri | Encyclopedia of Complexity and Systems Science[END_REF]. In what follows, we shall generalise time-translation-invariance in such a way that it becomes applicable to ageing phenomena. The hypothesis will be presented in section 2. In section 3 we shall show that not only the examples mentioned so far, but also several other well-established facts of ageing phenomenology follow from this single symmetry.

Generalised time-translation-invariance

Postulate: The Lie algebra generator X equi n of a time-space symmetry of an equilibrium system becomes a symmetry out of equilibrium by the change of representation

X equi n → X n = e ξ lnt X equi n e -ξ lnt ( 4
)
where ξ is a parameter whose value characterises the scaling operator φ on which X n acts. When applied to a dilatation generator X equi 0

= -t∂ t -1 z r∂ rδ , the prescription (4) leads to a modified effective scaling dimension δ eff = δξ , since

X equi 0 → X 0 = -t∂ t - 1 z r∂ r -δ -ξ (5a)
However, the time-translation generator

X equi -1 = -∂ t becomes X equi -1 → X -1 = -∂ t + ξ t (5b)
Whenever ξ = 0, time-translation-invariance is broken, but this is achieved here through the choice of a different representation rather than changing the Lie algebra of dynamical symmetries by suppressing X -1 . In general, it does not seem to be possible to require the presence of other symmetries, even if extended via the prescription (4), i.e. following ideas of local scale-invariance [22,23,[START_REF] Henkel | Non-equilibrium phase transitions vol. 2: ageing and dynamical scaling far from equilibrium[END_REF]. Such requirements, when applied to correlators, would in general imply that C = 0, because of certain Bargman superselection rules [41]. At most, one might hope that certain response functions R might admit larger dynamical symmetry groups [START_REF] Henkel | Non-equilibrium phase transitions vol. 2: ageing and dynamical scaling far from equilibrium[END_REF]. This subject however, must be postponed to a later date, since it is not the aim of this work to restrain the form of certain correlators or responses as much as possible by considering their co-variance under a maximal Lie algebra but rather to identify the minimal requirements on dynamical symmetries needed for an understanding of the general phenomenology of ageing. Generalised time-translations (5b) occurred first as a by-product of studies of local-scale-invariance for systems with dynamical exponent z = 2 [26]. They were advocated as a special case of much more general representations [39] in the context of holographic geometry of ageing. Physical examples arose only recently in the context of dynamical symmetries of biased systems [43,30,44].

The mathematical basis of the prescription (4) can be stated as follows. Proposition 1: [39,[START_REF] Henkel | [END_REF] For the constant γ and the non-constant function g(z), the generators

n = -z n+1 ∂ z -nγz n -g(z)z n ( 6
)
obey the conformal algebra [ n , m ] = (n -m) n+m for n, m ∈ Z.
Physically, the relevance of ξ for non-equilibrium dynamics can be understood from the kinetics of the spherical model [15,[START_REF] Godrèche | [END_REF]40,44]. This exactly solvable model can be formulated in terms of real-valued spin S(t, r) ∈ R subject to the con-straint ∑ r S 2 (t, r) = N , where N is the number of sites of the lattice. With the hamiltonian H [S] = -∑ (r,r ) S(t, r)S(t, r )µ ∑ r S 2 (t, r) the over-damped dynamics is given by ∂ t S(t, r) = -δ H [S] δ S(t,r) + z(t)S(t, r) + η(t, r), where η is a thermal white noise. Herein, the Lagrange multiplier z(t) is fixed from the spherical constraint. Defining ln g(t) := 2 t 0 dτ z(τ), it can be shown that this leads to a Volterra integral equation

g(t) = A(t) + 2T t 0 dτ f (t -τ)g(τ) (7)
for the temperature T where for nearest-neighbour interactions f (t) = e -4dt I 0 (4t) d and I 0 is a modified Bessel function. If d > 2, for a fully uncorrelated initial state [START_REF] Godrèche | [END_REF]. Hence, for d > 4 the model is in the mean-field universality class. On the other hand, whenever ξ = 0, fluctuation effects must be invoked for the required change of representation (4). On the other hand, if one prepares the system at T = T c (d) at critical equilibrium, the model merely undergoes equilibrium critical dynamics. Equilibrium initial conditions are spatially long-ranged and modify A(t) such that g(t) = cste. [40] and ξ = 0 throughout, for all d > 2. Therefore, representations with ξ = 0 constructed via eq. ( 4) should be relevant for non-equilibrium critical dynamics (where ageing occurs) and by extension also for phase-separation or phase-ordering kinetics, depending on whether the order-parameter is conserved or not, after a quench into the phase-coexistence region with T < T c . Our main use of dynamical symmetries will concern the scaling form of twopoint functions. A scaling operator φ a = φ a (t, r) is characterised by the two parameters (δ a , ξ a ). The co-variance of a two-point function

A(t) = f (t) such that at criticality with T = T c (d) one finds that g(t) ∼ t giving ξ = 2 = min(d/4 -1, 0)
C = C (t, s; r) = φ 1 (t, r)φ 2 (s, 0) (8)
under generalised time-translations (5b) and dilations (5a) leads to the conditions

X -1 C = -∂ t -∂ s + ξ 1 t + ξ 2 s C = 0 (9a) X 0 C = -t∂ t -s∂ s - 1 z r∂ r -δ 1 + ξ 1 -δ 2 + ξ 2 C = 0 (9b)
Eq. (9a) gives the meaning of 'broken time-translation-invariance' used in this work.

In what follows, all physical applications will taken from the following Proposition 2: The general solution of the co-variance conditions ( 9) is

C (t, s; r) = s -δ 1 -δ 2 +ξ 1 +ξ 2 t s -ξ 1 t s -1 -δ 1 -δ 2 +2ξ 1 F r (t -s) 1/z ( 10
)
where F is an undetermined scaling function.

Proof: Clearly, (10) solves both eqs. (9a,9b). General theorems on linear partial differential equations [START_REF] Kamke | Differentialgleichungen: Lösungsmethoden und Lösungen[END_REF] guarantee the uniqueness of this form, since a normalisation constant is absorbed into the undetermined function F .

3 Phenomenological consequences 3.1 Preliminaries: the ageing exponents b, a

For magnetic systems quenched from a disordered initial state to a temperature T ≤ T c , the two-time correlation function should obey dynamical scaling [7]

C(t, s; r) = κ bz C κ z t, κ z s; κr = s -b F C t s ; |r| s 1/z (11)
An important boundary condition is furnished by the equal-time correlator. If the system is quenched into the phase-coexistence region where T < T c , one expects

C(t,t; 0) ∼ 1 discrete (e.g. Ising model) M 2 eq continuous (12)
whereas for a quench onto the critical point T = T c , one should find

C eq (r) = C(∞, ∞; r) ∼ r -(d-2+η) (13)
and where η is an equilibrium critical exponent. These are captured by the asymptotic scaling function, for u 1

F C (1; u) ∼ u -bz ; b = 0 if T < T c (d -2 + η)/z if T = T c (14)
which furnishes the value of the ageing exponent b [7,[START_REF] Cugliandolo | Slow relaxations and non-equilibrium dynamics in condensed matter[END_REF]. For phase-ordering, with a non-conserved order-parameter and short-ranged interactions, one has z = 2 [7,8].

Similarly, for a response function one has [24,25,[START_REF] Henkel | Non-equilibrium phase transitions vol. 2: ageing and dynamical scaling far from equilibrium[END_REF]]

R(t, s; r) = s -1-a F R t s ; |r| s 1/z ; a = 1/z if class S (d -2 + η)/z if class L ( 15
)
where class S are systems with short-ranged equilibrium correlators which fall off exponentially fast and class L are systems with an algebraic long-range decay C eq (r) ∼ |r| -(d-2+η) and which includes all systems at criticality T = T c . Eq. ( 15) furnishes the value of the ageing exponent a [START_REF] Henkel | Non-equilibrium phase transitions vol. 2: ageing and dynamical scaling far from equilibrium[END_REF].

Auto-correlation function

The two-time auto-correlator C(t, s) = φ (t, r)φ (s, r) is built from two copies of the magnetic order-parameter φ . Because of the identitiy of the scaling operators, we have

δ = δ 1 = δ 2 , ξ = ξ 1 = ξ 2 (16)
such that the two-point function (10) becomes, in the limit of large separations y = t/s 1

C(t, s) = s -2δ +2ξ t s -2δ +ξ F C (0) (17)
This does indeed agree with the expections (2,3) and in particular proves the expected algebraic asymptotics (3a) of the scaling function f C (y). We can identify

if T = T c : b = 2(δ -ξ ) = d -2 + η z , λ C z = 2δ -ξ = ξ + d -2 + η z (18a) if T < T c : b = 2(δ -ξ ) = 0 , λ C z = 2δ -ξ = ξ (18b)
where we used (14).

Auto-response function

The auto-response function R(t, s) = φ (t, r) φ (s, r) can be written as a correlator of the order-parameter φ with the conjugate response scaling operator φ . Furthermore, since responses are indeed co-variant under larger algebras of local scaletransformations, notably conformal transformations which make up local scaleinvariance [22,23,[START_REF] Henkel | Non-equilibrium phase transitions vol. 2: ageing and dynamical scaling far from equilibrium[END_REF], we now have

δ = δ 1 = δ 2 , ξ = ξ 1 , ξ = ξ 2 (19)
where the first identity is a consequence of local-scale invariance. However, the two parameters ξ and ξ are now independent. Then, for y = t/s 1 we have from ( 10)

R(t, s) = s -2δ +ξ + ξ t s -2δ +ξ F R (0) (20)
once more in agreement with (2,3a). We can identify, for both classes S and L

1 + a = 2δ -ξ -ξ , λ R z = 2δ -ξ (21)
The results obtained so far can be summarised as follows. 1. For large arguments y 1, both scaling functions are algebraic

f C (y) ∼ y -λ C /z , f R (y) ∼ y -λ R /z (22)
2. Comparison of ( 18) and ( 21) implies the exponent equality λ C = λ R .

Hence both properties (3) find their natural explanation as a consequence of generalised time-translation-invariance (9a), combined with dynamical scaling.

Finite-size effects

If the ageing system is placed in a volume of linear size N, then the shape of the auto-correlator C(t, s; N -1 ) is modified in a characteristic way which is schematised in figure 2, here for the case of phase-ordering kinetics at T < T c , where also b = 0. For reasonably large, but not too large, values of y, the autocorrelator's behaviour is well approximated by the infinite-size behaviour C ∞ (ys, s) = C(ys, s; 0) ∼ y -λ /z (dashed line in figure 2). However, if y is increased further, the data first decay more rapidly and then saturate at a y-independent plateau

C (2) ∞ = lim y→∞ C ys, s; 1 N (23)
Can one understand this behaviour in terms of a scaling description based on (4) ? How does the plateau height C

∞ depend on either N or the waiting time s ? Finitesize effects are also studied experimentally [51,52].

According to the theory of finite-size scaling [START_REF] Fisher | Critical phenomena[END_REF][START_REF] Barber | Finite-size scaling[END_REF][START_REF] Suzuki | [END_REF], in a finite volume the inverse size 1/N becomes a further relevant variable. Therefore, the two-time autocorrelator (with spatial separation r = 0) should obey the two co-variance conditions

X -1 C = -∂ t -∂ s + ξ t + ξ s C = 0 (24a) X 0 C = -t∂ t -s∂ s - 1 z 1 N ∂ 1/N -2 δ -ξ C = 0 (24b)
which have the unique solution (re-using eqs. ( 18) and (3b), of course) for t s C t, s;

1 N = s -b t s -λ /z F C N t 1/z (25)
The interpolation between systems of infinite and finite volume, respectively, is controled by the finite-size scaling function F C (u), of the single argument u = Nt -1/z . Clearly, for large arguments F C (∞) = cste., such that the infinite-size behaviour (3) is recovered. On the other hand, for u 1, one should have F C (u) ∼ u -λ such that the plateau is indeed y-independent. This implies for the plateau height C

(2) ∞ ∼ s λ /z-b N -λ . The requirement (24a) is essential to reduce the scaling function F C to a function of a single variable on which the traditional finite-size scaling arguments [START_REF] Fisher | Critical phenomena[END_REF][START_REF] Barber | Finite-size scaling[END_REF] can be applied. Of course, one might as well assemble heuristic arguments for the scaling C

(2) ∞ ∼ N -λ for s fixed, see e.g. [29].

Proposition 4: For quenches to T ≤ T c , the plateau height C

(2) ∞ of the autocorrelator in a finite volume of linear size N obeys the scalings if the waiting time s is fixed: C (2) ∞ ∼ N -λ (26a)

if the system size N is fixed: C (2) ∞ ∼ s λ /z-b (26b)
where the values (18) must be used for the exponent b.

For a test of ( 26) in the spherical model at T < T c , see [29]. Tests in the 2D Glauber-Ising model are forthcoming [49]. In practise, the determination of λ from simulations and/or experiments may become difficult through the need to study sufficiently large waiting times and also y = t/s large enough such that the asymptotics f C (y) ∼ y -λ /z can be observed. This requires to create data for a sufficiently wide window. Eq. ( 26) rather suggests to let the auto-correlator relax to the plateau and extract λ from the scaling of C

(2) ∞ which may be the better computational tool [49]. Since in experiments [2], N is likely fixed from the sample preparation techniques, eq. (26b) might be an useful alternative for the determination of λ /z.

Analogous results can be easily derived for instantaneous responses R(t, s; r), or their time-integrated counterparts χ = du R(t, u), for systems in a finite volume.

Finally, the characteristic length scale L(t) may for example be extracted as a second moment of the single-time correlator C(t; r), using (25) as follows which gives a method to measure the dynamic exponent z. This finite-size crossover has been routinely observed in numerical simulations of lattice systems [10,11,12,13,31].

L 2 (t) = V dr r 2 C(t; r; N -1 ) V drC(t; r; N -1 ) = t 2/z F L N t 1/z ∼ t 2/z volume infinite N 2 volume finite (27)

Magnetic initial states

For a quench onto criticality T = T c , the magnetisation m(t) has a non-trivial timeevolution [33] sketched in figure 3. If the system is prepared with an initial magnetisation m(0) = m 0 = 0 but is otherwise uncorrelated, there exists a short-time critical regime where m(t) ∼ t Θ where the slip exponent Θ is independent 6 of the equilibrium critical exponents and z. Its value can be found from deep field-theoretic considerations based on the renormalisation group and short-time operator product expansions [33,[START_REF]From phase transitions to chaos[END_REF]. This leads to the celebrated JANSSEN-SCHAUB-SCHMITTMANN (JSS) scaling relation [33]

Θ = d -λ z ( 28
)
In this formulation, the identity (3b) between auto-correlation and auto-response exponents was used. After passing through a maximum, the magnetisation crosses over into a regime of critical decay, according to m(t) ∼ t -β /(νz) , where β and ν are standard equilibrium critical exponents [1,[START_REF] Täuber | Critical dynamics: a field-theory approach to equilibrium and non-equilibrium scaling behaviour[END_REF]. Can one understand this behaviour and the existence of two distinct scaling regimes in figure 3, in terms of a scaling description based on (4) ?

We shall first relate the behaviour of m(t) to a response function. In order to see how this comes about, recall from Janssen-de Dominicis non-equilibrium fieldtheory [START_REF] De Dominicis | [END_REF]32,[START_REF]From phase transitions to chaos[END_REF][START_REF] Täuber | Critical dynamics: a field-theory approach to equilibrium and non-equilibrium scaling behaviour[END_REF]

the calculation of averages of an observable

A A = Dφ D φ A[φ ]e -J [φ , φ ] (29)
where the action

J [φ , φ ] = J b [φ , φ ] + J ini [ φ ]
is decomposed into a bulk term and an initial term

J b [φ , φ ] = dtdr φ ∂ t -∆ r -V [φ ] φ -T φ 2 (30a) J ini [ φ ] = dr 1 2τ 0 φ 2 (0, r) -m 0 φ (0, r) (30b)
(with the usual re-scalings) which describes in the bulk a thermal white noise of temperature T , and an initially magnetised state of average magnetisation m 0 together with gaussian fluctuations of width τ 0 . Herein, 'initial time' means of course a timescale τ mic at the beginning of the scaling regime. The bulk action at temperature

T = 0, J 0 [φ , φ ] = lim T →0 J b [φ , φ
] is called the deterministic action. Deterministic averages • 0 are obtained analogously to (29) where the action J is replaced by its deterministic part J 0 . Either from causality considerations [33,[START_REF]From phase transitions to chaos[END_REF] or localscale-invariance [41] one obtains the Bargman superselection rules

n times φ • • • φ m times φ • • • φ 0 ∼ δ n,m (31)
for the deterministic averages. This means that only observables built from an equal number of order-parameters φ and conjugate response operators φ can have nonvanishing averages.

Averages should now be computed from (29). They can be reduced to deterministic averages by formally expanding in powers of T, 1 τ 0 and m 0 to all orders. It follows that the time-dependent magnetisation can be obtained as

m(t) = φ (t, 0) = m 0 R d dr φ (t, 0) φ (0, r) 0 = m 0 R d dr R(t, 0; r) = m 0 R(t, 0; 0) (32)
This follows from (29) and the form of the action J [φ , φ ] detailed in (30). Namely, the only term admissible, because of the Bargman rule (31), is the one linear in m 0 as can be seen from (30b). By definition, this is a response function and in the last step, we introduced the spatial Fourier transform

R(t, s; q) = R d dr e -iq•r R(t, s; r) (33)
Eq. ( 32) is the starting point of our scaling analysis. Again, the formally vanishing waiting time s = 0 in (32) should be physically interpreted as a microscopically small time s mic t at the beginning of the scaling regime.

In the physical situation at hand, the initial magnetisation m 0 should be considered as a further dimensionful scaling variable, with scaling dimension x 0 . The co-variance conditions of the global reponse function R = R(t, s; 0; m 0 ) then read

X -1 R = -∂ t -∂ s + ξ t + ξ s R = 0 (34a) X 0 R = -t∂ t -s∂ s - x 0 z m 0 ∂ m 0 -2δ + d z + ξ + ξ R = 0 (34b)
In analogy to what was done before, we have for t/s 1 the asymptotics R(t, s; 0; m 0 ) = s -2δ +d/z+ξ + ξ t s

-2δ +d/z+ξ F R m 0 t -x 0 /z (35)
with the scaling function F R (u) of the single argument u = m 0 t -x 0 /z . Since we need the response with respect to an 'initial' perturbation, we must now send s → s mic and then can absorb it into the scaling function. Combination with (32) gives for the sought time-dependent magnetisation

m(t) = m 0 t Θ F R m 0 t -x 0 /z , Θ = d z + ξ -2δ = d -λ z (36)
Herein, we identified the slip exponent and then observe that we indeed reproduce the JSS scaling relation [START_REF] Henkel | [END_REF]. The scaling function F R interpolates between the two regimes of non-equilibrium critial scaling, through the assumed properties (i) F R (0) = cste. which reproduces the short-time scaling regime and (ii) F R (u) ∼ u -1 for u 1 which makes the late-time scaling regime independent of m 0 and which also fixes x 0 .

Proposition 5: A critical system, with initial magnetisation m 0 but otherwise uncorrelated, has the time-dependent magnetisation

m(t) = m 0 t Θ F R m 0 t Θ +β /(νz) ( 37
)
where Θ is given by [START_REF] Henkel | [END_REF] and β , ν are standard equilibrium critical exponents.

Hence the qualitative behaviour of figure 3 is reproduced. Given the generic relationship (32), the derivation of the JSS scaling relation [START_REF] Henkel | [END_REF] reduces to a simple Fourier transform. Since the scaling function stems from a response function, it is conceivable that larger dynamical symmetries such as local scale-invariance might be brought to furnish further details on its behaviour. We hope to return to this elsewhere.

This result should only be applicable to critical quenches. Any attempt to prepare an initially magnetised state should lead for T < T c to a rapid relaxation towards one of the several distinct equilibrium states in the phase-coexistence region such that the system will leave the regions in parameter space where dynamical scaling holds.

Global correlators

While all observables discussed so far have been local correlators or responses, we now consider global quantities. For example, in an Ising model with spins σ n = ±1 at each site n of a lattice Λ ⊂ Z d , the global spin-spin correlator is defined as

Q(t, s) := 1 |Λ | ∑ n,m∈Λ σ n (t)σ m (s) ( 38
)
where Λ is the number of sites of the lattice. Analogous definitions apply to other spin models. What can be said on the long-time behaviour of such observables ?

Proposition 6: For a magnetic system quenched to T ≤ T c , the global spin-spin correlator (38) obeys for large observation times t

Q(t, 0) ∼ t Θ (39)
where the slip exponent Θ is given by [START_REF] Henkel | [END_REF]. Proof: At criticality, the result has been known since a long time [START_REF] Tomé | [END_REF]. It is derived here for all T ≤ T c as a further example of generalised time-translation-invariance (5b). Taking into account spatial translation-invariance, in the continuum limit the discrete sums over lattice sites are replaced by integrations over a spatial domain V ⊂ R d . Sending the integration volume |V | to infinity, one has

Q(t, s) = lim |V |→∞ 1 |V | V ×V drdr φ (t, r)φ (s, r ) = R d dr C(t, s; r) = C(t, s; 0)
where the definition (33) of the Fourier transform was used. Recall the scaling forms (11,[START_REF] De Dominicis | [END_REF] and carry out the spatial integration. This gives for the t s asymptotics

C(t, s; 0) = s -b+λ /z t -λ /z+d/z R d du F C (|u|) = cste. ∼ s -Θ +(2-η)/z t Θ if T = T c s -Θ +d/z t Θ if T < T c (40)
where the scaling forms (2,[START_REF] De Dominicis | [END_REF] and the identifications (18,[START_REF] Majumder | [END_REF] were used, which relate the slip exponent Θ with the autocorrelation exponent λ . We point out that this establishes the validity of the identity (28) beyond the realm of non-equilibrium critical dynamics for which it was originally derived [33]. Finally, the mathematical limit s → 0 corresponds physically to the limit s → s mic , where s mic is a microscopic time-scale for the onset of the dynamic scaling regime. Hence the dependence on s mic can be absorbed into the scaling function

F C . Then Q(t, 0) ∼ Q(t, s mic ) ∼ t Θ as asserted.
For non-equilibrium critical dynamics, eq. ( 39) has been routinely used to measure the exponents Θ (or, via [START_REF] Henkel | [END_REF], equivalently the autocorrelation exponent λ , if z is known) [START_REF] Tomé | [END_REF]1,42]. The derivation given here makes it clear that eq. ( 39) along with the scaling relation ( 28) furnishes an alternative method to find λ from simulations for quenches into the low-temperature phase T < T c . We are not aware of any numerical work along this line.

For estimating the dynamical exponent z, computing L(t) from a second moment is tedious. An often-used alternative relies on the scaling of the single-time correlator, e.g. for phase-ordering where b = 0 one may look for pairs (t, L(t)) by solving numerically a scaling equation such as C(t, r) = F C (|r|/L(t)) = 0.5 [START_REF] Majumder | [END_REF]37]. Yet a further alternative studies the short-time dynamics of the squared magnetisation m 2 (t) [31]. Repeating the above arguments, notably eq. ( 40), it is easily seen that

m 2 (t) ∼ C(t,t; 0) ∼ t d/z-b ∼ t d/z if T < T c t (2-η)/z if T = T c (41)
and in agreement with earlier heuristic arguments [START_REF] Tomé | [END_REF]42,31]. The numerical requirements of this technique are somewhat smaller than in the other methods mentioned above. In recent applications to phase-ordering kinetics in the 2D Ising model it was concluded that as a tool for numerical computation the accuracy of ( 41) is at least equal to the one of other techniques [31].

Proposition 7: In magnetic systems quenched to T ≤ T c , the time-dependent averaged squared magnetisation scales as

m 2 (t) ∼ t d/z-b = t d/z if T < T c [31] t (2-η)/z if T = T c [48] ( 42
)
where η is a standard equilibrium critical exponent.

The extension of Propositions 6 and 7 to systems in a finite volume is obvious.

Conclusions

The phenomenology of simple physical ageing is based on several expectations which are met in almost all systems and models under study. However, these auxiliary assumptions have always been presented as being independent of each other. The point of this work is to provide an unifying perspective such that a single hypothesis is sufficient from which the entire generic phenomenology of simple ageing can be derived. As stated in eq. ( 4), non-equilibrium dynamics distinguishes itself from equilibrium dynamics merely through a different choice of Lie algebra representation. We have considered here an intertwining operator which only depends on time such that merely the time-translation and dilation generators change their form. In Propositions 3 to 7 we showed how the entire known phenomenology of the simple ageing at T ≤ T c of local and global correlators and responses can be reproduced: in spatially infinite or finite systems or in systems with or without an initial magnetisation. Simple proofs of the JSS scaling relation [START_REF] Henkel | [END_REF] were given. Exponent values are not predicted. Herein, it is essential that time-translation-invariance is broken 'softly' through time-dependent terms in its Lie algebra generator X -1 , rather than leaving it out altogether, as it was usually done up to now. In this way, the observables contain at most a scaling function of a single variable, which allows the application of standard phenomenological scaling arguments.

Complementary hypothesis needed to carry out this programme include:

1. simple ageing with an algebraically growing length scale L(t) ∼ t 1/z was admitted throughout. 2. all initial correlations were assumed short-ranged. Otherwise, results such as λ C = λ R need no longer hold true [7,[START_REF] Henkel | Non-equilibrium phase transitions vol. 2: ageing and dynamical scaling far from equilibrium[END_REF]. 3. attention was restricted to the sole intertwining operator W = e -ξ lnt , see eq. ( 4).

Future work should relax or further clarify at least some of these assumptions.

Fig. 1

 1 Fig. 1 Schematic free energies. The circle represents the disordered initial state.

Fig. 2 Proposition 3 :

 23 Fig. 2 Scaling of the autocorrelator C(ys, s; N -1 ) in a finite volume (with b = 0).

Fig. 3

 3 Fig. 3 Time-dependence of the non-equilibrium magnetisation m(t).

For simplicity, we shall use the language of classical dynamics, although all considerations which follow should apply to quantum quenches as well.

Hence glassy systems where L(t) ∼ lnt 1/ψ grows logarithmically[19, 53,

3] are not considered.

This explicitly excludes multi-scaling as found e.g. in phase-separation kinetics at T < T c with conserved order-parameters[14,9], as well as logarithmic sub-ageing[START_REF] Berthier | [END_REF]18].

Spatial translation-invariance as well as spatial rotation-invariance such that r → r = |r| will be admitted throughout for notational simplicity.

At criticality, this may be argued from the finiteness of the limit fluctuation-dissipation ratio[START_REF] Godrèche | [END_REF] X ∞ = lim y→∞ lim s→∞ X(ys, s) with X(t, s) = T C R(t, s; 0) ∂ s C(t, s; 0) . This argument does not apply to T < T c , but for short-ranged and gaussian initial correlators, (3b) still follows, either from a comparison of the scaling form of correlators and responses[6,7] or else, if in addition z = 2, from local scale-invariance[41,[START_REF] Henkel | Non-equilibrium phase transitions vol. 2: ageing and dynamical scaling far from equilibrium[END_REF].

Time-translation-invariance & ageing

In mean-field theories, Θ = 0. Both Θ > 0 and Θ < 0 is seen in models, see[START_REF] Henkel | Non-equilibrium phase transitions vol. 2: ageing and dynamical scaling far from equilibrium[END_REF].

Remerciements This work was supported by the french ANR-PRME UNIOPEN (ANR-22-CE30-0004-01) and PHC Rila (51305UC).